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of the sciences and arithmetic the queen of mathe- 
matics. She often condescends to render service 
to astronomy and other natural sciences, but under 
all circumstances the first place is her due." 

So wrote the master mathematician, astron- 
omer, and physicist, Gauss (1777-1855) over a 
century ago. Whether as history or prophecy, 
Gauss' declaration is far from an overstatement. 
Time after time in the nineteenth and twentieth 
centuries, major scientific theories have come into 
being only because the very ideas in terms of 
which the theories have meaning were created 
by mathematicians years, or decades, or even 
centuries before anyone foresaw possible applica- 
tions to science. 

Without the geometry of Riemann, published 
in 1854 X or without the theory of in variance de- 
veloped by the mathematicians Cay ley (1821- 
1895), Sylvester (1814-1897), and a host of their 
followers, the general theory of relativity and 
gravitation of Einstein in 1916 could not have 


been stated. Without the whole mathematical 
theory of boundary value problems to use a 
technical term which need not be explained now 
originating with Sturm (1803-1855) and Liouville 
(1809-1882), the far-reaching wave mechanics of 
the atom of the past five years would have been 

The revolution in modern physics which began 
with the work of W. Heisenberg and P. A. M. 
Dirac in 1926 could never have started without 
the necessary mathematics of matrices invented 
by Cayley in 1858, and elaborated by a small 
army of mathematicians from then to the present 

The concept of invariance, of that which 
remains unchanged in the ceaseless flux of nature, 
permeates modern physics, and it originated in 
1801 in the purely arithmetical work of Gauss. 

These are but a few of many similar instances. 
In none of the scores of anticipations of fruitful 
applications to science was there any thought of 
what might come out of the pure mathematics. 
Guided only by their feeling for symmetry, 
simplicity, and generality, and an indefinable 
sense of the fitness of things, creative mathe- 
maticians now as in the past are inspired by the 
art of mathematics rather than by any prospect 
of ultimate usefulness. However it may be in 


engineering and the sciences, in mathematics the 
deliberate attempt to create something of imme- 
diate utility leads as a rule to shoddy work of only 
passing value. The important practical and scien- 
tific applications of mathematics are unsought 
byproducts of the main purposes of professional 

The queen of the sciences however needs no 
shabby apology as an introduction. Jacobi 
(1804-1851) fittingly expressed what many be- 
lieve to be the true purpose of mathematics in 
his retort to Fourier (1768-1830). To appreciate 
this we must recall that Fourier's influence on 
pure mathematics is comparable to Jacobi's on 
applied mathematics. In his analytical theory of 
heat (published in 1822), the applied mathe- 
matician Fourier devised tools which are as useful 
today in pure mathematics as they are in all 
physics where wave motion underlies the pattern 
of events. On the other hand, the contributions 
of the pure mathematician Jacobi to higher me- 
chanics are indispensable in modern physics. 
Fourier had reproached Jacobi for "trifling with 
pure mathematics." Jacobi replied that a scientist 
of Fourier's calibre should know that the true 
end of mathematics is the greater glory of the 
human mind. 


In the past hundred years mathematics entered 
its golden age. This most prolific period in the 
history of mathematics had well started by 1830; 
the end is not yet in sight. No previous age ap- 
proaches the past century for the depth and 
tremendous sweep of its mathematics. The only 
other centuries at all comparable with the past 
hundred years are those of Archimedes (287-212 
B.C.) and Newton (1642-1727), and these can be 
compared with the Century of Progress only 
when generous allowance is made for the diffi- 
culties of pioneering. The mathematical inheri- 
tance of the past century from its predecessors 
was great, both in quantity and quality, so great 
indeed that one prophet in 1830 lamented that 
"the golden age of mathematical literature is un- 
doubtedly past." That splendid inheritance of 
at least twenty centuries was increased many 
times in one hundred years. 

So vast has been the increase of mathematical 
knowledge in the past century that few men would 
presume to claim more than an amateur's ac- 
quaintance with more than one of the four major 
divisions of modern mathematics. The field of 
higher arithmetic alone as it exists today is prob- 
ably beyond the complete mastery of any two 


men, while geometry, algebra and analysis, es- 
pecially the last, are of even greater extent. If 
mathematical physics be annexed as a province of 
mathematics, a detailed, professional mastery of 
the whole domain of modern mathematics would 
demand the lifelong toil of twenty or more richly 
gifted men. 

In all this there is a crumb of comfort for those 
whose mathematical training ended with their 
last year in high school or their first year in 
college. These are not so much worse off, rela- 
tively, than the majority of mathematicians who 
turn the pages of the current mathematical 
periodicals or attend scientific meetings. Out of 
fifty mathematical papers presented in brief at 
such a meeting, it is a rare mathematician in- 
deed who really understands what more than 
half a dozen are about. The very language in 
which most of the other forty-four are presented 
goes clean over the head of the man who follows 
the six reports nearest his own specialty./ 

Many causes contribute to this state of affairs 
which seems to be a necessary consequence of 
mathematical progress. We need mention only 
one. It is the perennial youthfulness of mathe- 
matics itself which marks it off with a discon- 
certing immortality from the sciences. 

In theoretical physics it is but seldom neces- 


sary to master in detail a work published over 
thirty years ago, or even to remember that such a 
work was ever written. But in mathematics the 
man who is ignorant of what Pythagoras said in 
Croton in 500 B.C. about the square on the long- 
est side of a right angled triangle, or who forgets 
what someone in Czecho Slovakia proved last 
week about inequalities, is likely to be lost. The 
whole terrific mass of well established mathe- 
matics, from the ancient Babylonians to the 
modern Japanese, is as good today as it ever was. 

Looking down and far out over the past from 
our vantage points of today we can only marvel 
at the dogged courage and persistence of the 
explorers who first won a devious way through 
the wilderness. Broad highways now cross the 
barren deserts, straight as bowstrings; where 
scores perished miserably in the pestilent marshes 
there is a thriving city, and the pass through the 
iron mountains which our forefathers sought in 
vain is an easy four hours' pleasure trip from the 
distant city. The loftier range behind the one on 
which we stand is now accessible to us, although 
the way is hard, and by scaling its lesser peaks 
we can catch glimpses of an El Dorado of which 
the most daring of the pioneers never dreamed. 

If we marvel at the patience and the courage of 
the pioneers, we must also marvel at their per- 


sistent blindness in missing the easier ways through 
the wilderness and over the mountains. What 
human perversity made them turn east to perish 
in the desert, when by going west they could 
have marched straight through to ease and plenty? 
This is our question today. If there is any con- 
tinuity in the progress of mathematics, the men of 
a hundred years hence will be asking the same 
question about us. We know that there is a 
higher range behind us, and we suspect that 
behind that one is a higher, and so on, for as far 
and as long as there shall be human beings with 
the spirit of adventure to heed the whisper of the 
unknown. At the present rate of progress our 
vantage points of today will be barely distinguish- 
able hillocks in a boundless plain to the explorers 
of a century hence. Before standing on one or 
two of the hard-won peaks of the past century to 
see what we can of the progress made in the last 
hundred years, let us look about us well before we 


To get some sort of a perspective, let us con- 
sider roughly the kind of mathematics acquired 
by a student who takes all that is offered in a 
good American high school. The geometry taught 
is practically that of Euclid and is 2200 years old. 


It is a satisfactory first approximation to the 
geometry of the physical universe, and it is good 
enough for engineers, but it is not that which is 
of vital interest in modern physics, and its interest 
for working mathematicians evaporated long ago. 
Our vision of the universe has swept far beyond 
the geometry of Euclid. 

In algebra the case is a little better. A well 
taught student will master the binomial theorem 
for a positive whole number exponent which 
Blaise Pascal discovered in 1653. There he will 
stop. And yet the really interesting things in 
algebra are the creation of the Nineteenth and 
Twentieth Centuries, and began to be developed 
over a century and a half after Pascal died. 

Of higher arithmetic the graduate of a good 
school will learn precisely nothing. Unless ex- 
tremely fortunate, he will never even have heard 
of the theory of numbers. And yet at least one 
of its most beautiful and far reaching truths was 
known to Euclid. Many of the most striking 
results in this field are accessible to anyone with 
a year of high school training. 

In analytical geometry and the calculus the 
score is again zero. The calculus, however, which 
has been estimated as the most powerful instru- 
ment ever devised for scientific thought, may 


become part of the regular high school course 
before the next Chicago World's Fair. 

Without a good working knowledge of the 
differential and integral calculus created by 
Newton and Leibnitz in the Seventeenth Century 
it is impossible even to read serious works on the 
physical sciences and their applications, much 
less to take a step ahead. The like is true, but to 
a far lesser extent, for some branches of biology 
and psychology, and it is beginning to be true for 
some economics. Any normal boy or girl of sixteen 
could master the calculus in half the time often 
devoted to stumbling through Book One of 
Caesar's Gallic War. And it does seem to some 
modern minds that Newton and Leibnitz were 
more inspiring leaders than Julius Caesar and 
his unimaginative lieutenant Titus Labienus. 

The junior college student will be considerably 
farther ahead at the end of his fourth year. 
Provided he has not sought culture by the literary 
trail exclusively, he may be able to appreciate 
some of the minor classics of science. He will 
know as much as the men of the Eighteenth Cen- 
tury knew of the calculus, and he will know it 
better than they did. Much of what passed for 
proof with the pioneers would not now be tolerated 
in a college text book. To this light extent the 
profound critical work of the Nineteenth Century 


mathematicians has influenced the thinking of 
those who take the calculus in college at least in 
a good college under a man who is not hopelessly 
dry and dusty. 

Before quitting this somewhat uninspiring pros- 
pect, let us glance at another of the reasons why 
the average graduate of standard four year college 
course in mathematics usually manages to miss 
completely the spirit of modern mathematics. 
The point is obvious from a remark of Abel 
(1802-1829), one of the greatest mathematical 
geniuses of all time. In his wretched life of less 
than twenty-seven years Abel accomplished so 
much of the highest order that one of the leading 
mathematicians of the Nineteenth Century 
(Hermite, 1822-1901) could say without exaggera- 
tion, "Abel has left mathematicians enough to 
keep them busy for five hundred years." Asked 
how he had done all this in the six or seven years 
of his working life, Abel replied "By studying the 
masters, not the pupils." 

To appreciate the living spirit rather than the 
dry bones of mathematics it is necessary to inspect 
the work of a master at first hand. Text books 
and treatises are a very necessary evil. The mere 
bulk of the work to be assimilated in any reason- 
able time precludes intimate contact with the 
creators through their works. Nevertheless it is 


not impossible in the ordinary course of education 
to read at least ten or twenty pages of mathe- 
matics as it came from the pen of a master. The 
very crudities of the first attack on a significant 
problem by a master are more illuminating than 
all the pretty elegance of the standard texts which 
has been won at the cost of perhaps centuries of 
finicky polishing. 

It is rarely feasible for beginners to attempt 
the mastery of recent work. This appears in the 
mathematical journals, of which there are now 
about 500 published throughout the world. Some 
come out monthly, others quarterly, and the con- 
tents of about 200 are almost exclusively accounts 
of current mathematical research. Most of the 
articles are in English, Italian, French, or German 
(particularly the last two), although many are 
printed in the native languages of the authors, 
which range from Japanese, Russian, and Polish to 
Czech and Roumanian. For a competence in 
modern mathematics a reading knowledge of the 
first four languages named is a necessity. 

Instead of trying to touch the spirit of modern 
mathematics through any of this up-to-the-minute 
work, it is much more practicable to study 
attentively some older classic. Many of the 
fluent papers of Euler (1707-1783), for example, 
dealing with quite elementary things, may be read 


as easily as a detective thriller. A little farther 
along, a memoir by Lagrange (1736-1813) would 
make an excellent companion to all the clumsy 
textbooks of the standard college course in ana- 
lytical mechanics. 

In this connection there is an amusing bit of 
recent history. At one of the leading American 
universities the ambitious president had so thor- 
oughly grasped Abel's precept about studying the 
masters in preference to their pupils, that he pro- 
ceeded to put it into effect in the freshman class. 
To aid him in this worthy undertaking, the presi- 
dent called in a specialist in the teaching of 
science not a specialist in science. Between them 
they made up a list of mathematical classics to 
be read by freshmen in their spare time. These 
included Newton's Principia of 1687 and 
Einstein's Theory of General Relativity and Gravi- 
tation of 1916. The last is quite a short trifle. 
It is the famous paper of which it used to be said 
that only twelve men in the world could under- 
stand it. The president is enthusiastic about the 
project. The freshmen have not yet been heard 


None of these remarks on the antiquity of the 
mathematics which passes as sufficient in a liberal 


education today, or on Abel's sound advice to 
would-be mathematicians, are intended in any 
spirit of discouragement. Quite the reverse: by 
admitting that it is a waste of time for those who 
are not mathematicians by trade to explore the 
minutiae of modern mathematics, we shall agree 
to be content with wider vistas than would satisfy 
a peering professional. In fact one of the out- 
standing achievements of the past century was the 
discovery and exploration of loftier points of view 
from which many fields of mathematics, both 
ancient and modern, can be seen as wholes and 
not as rococo patchworks of dislocated special 
problems. The details, however, remain matters 
which only specialists can appreciate. 

The summits from which those broader points of 
view may be gained today seem to us, who did 
not have the pain of discovering them, to be 
ridicuously evident. Why were these outstand- 
ing peaks not seen before? Viewing the progress 
of mathematics one might almost be tempted to 
amend Kant's rhapsody, 

"Two things I contemplate with ceaseless awe, 
The starry Heavens, and man's sense of law/' 

by striking out "sense of law" and substituting for 
it "stupidity." The only thing that deters us is 
the moral certainty that we ourselves are as blind 


to what stares us in the face as our predecessors 

If the mathematical spirit of the past hundred 
years can be described in a phrase, probably 
ever greater generality and ever sharper self-criticism 
is as just as any. Interest in special or isolated 
problems steadily diminished as the century ad- 
vanced, and mathematicians became builders of 
vast and comprehensive systems of knowledge in 
which individual theorems were completely sub- 
ordinated to the grander structure of inclusive 
theories. The fashioning of ever more powerful 
weapons for the assault of whole armies of old 
difficulties, instead of single combat against one 
at a time, also characterized this golden age of 
mathematics. And through and over the whole 
period played an almost continuous brilliance of 
the most amazing inventiveness the world has 
ever known. 

The other side of the picture is increasing rigor. 
The so-called obvious was repeatedly scrutinized 
from every angle and was frequently found to be 
not obvious but false. "Obvious" is the most 
dangerous word in mathematics. 




WHATEVER mathematics 
was a century ago, it is certainly not today the 
meagre shadow of itself which the dictionaries 
make of it. No doubt it takes courage amounting 
to rashness to quarrel with a standard dictionary, 
but mathematicians have never been conspicuous 
for that particular brand of cowardice which sub- 
mits to the printed word merely because it is fat, 
black, and backed by authority. Disregarding 
tradition, some have even framed pithy definitions 
of their own, intended as improvements on those 
of the dictionaries. 

Unfortunately no two of the definitions are in 
complete agreement. Each has some high light 
which reflects the bias of its author, and all taken 
together might give an impressionistic picture not 
utterly inadequate. To reproduce all of these 
attempts to hobble mathematics in a neat phrase 
would amount to compiling a mathematical dic- 
tionary, and the work would be hopelessly out of 
date long before it was finished. A few examples 
must suffice. 



The first description of mathematics as a whole 
which need be seriously considered is a much- 
quoted epigram which Bertrand Russell emitted 
in 1901. 

"Mathematics may be defined as the subject 
in which we never know what we are talking 
about, nor whether what we are saying is true." 

This has four great merits. First, it shocks the 
self-conceit out of common sense. That is pre- 
cisely what common sense is for, to be jarred into 
uncommon sense. One of the chief services which 
mathematics has rendered the human race in the 
past century is to put 'common sense' where it 
belongs, on the topmost shelf i]text to the dusty 
canister labled 'discarded nonsense.' 

Secondly, Russell's description emphasizes the 
entirely abstract character of mathematics. 

Thirdly, it suggests in a few words one of the 
major projects of mathematics during the past 
half century, that of reducing all mathematics and 
the more mature sciences to postulational form 
(which will be explained later), so that mathe- 
maticians, philosophers, scientists and men of 
plain common sense can see exactly what it is 
that each of them imagines he is talking about. 

Last, Russell's description of mathematics ad- 
ministers a resounding parting salute to the 
doddering tradition, still respected by the makers 


of dictionaries, that mathematics is the science of 
number, quantity, and measurement. These 
things are an important part of the material to 
which mathematics has been applied. But they 
are no more mathematics than are the paints in an 
artist's tubes the masterpiece he paints. They 
bear about the same relation to mathematics that 
oil and ground ochre bear to great art. 

Although it is true in a highly important sense, 
of which examples will appear as we proceed, 
that we do not know what we are talking about in 
mathematics, there is another side to the story, 
which distinguishes mathematics from the elusive 
reasoning of some philosophers and speculative 
scientists. Whatever it may be that we are 
talking about in a mathematical argument, we must 
stick to the subject and avoid slipping new assump- 
tions or slightly changed meanings into the things 
from which we start. 

To be certain that we have not shifted the sub- 
ject of discussion in an involved and delicate 
mathematical argument, or to know that our 
initial assumptions really do contain all that we 
think we are talking about, is the crux of the whole 
matter. Time and again mathematicians have 
been forced to tear down elaborate structures of 
their own building because, like any other fallible 


human beings, they have overlooked some trivial 
defect in the foundations. 

Before leaving Russell's definition, let us put 
two others beside it for comparison. According 
to Benjamin Pierce (1809-1880), "Mathematics 
is the science which draws necessary conclusions." 
As Russell restates the same idea, "Pure mathe- 
matics consists entirely of such asseverations as 
that, if such a proposition is true of anything, 
then such and such another proposition is true. 
It is essential not to discuss whether the first 
proposition is really true, and not to mention what 
the anything is of which it is supposed to be true." 
Or again, "Pure Mathematics is the class of all 
propositions of the form 'p implies q/ where p, q 
are propositions . . . . " 

The evolution of this excessively abstract view 
of mathematics has been slow, and it is a charac- 
teristic product of mathematical activity of the 
past half century. Not all mathematicians would 
assent to a definition of this type. Many, par- 
ticularly those of the older generation, prefer 
something more concrete. 

These estimates may well be enhanced by one 
from Felix Klein (1849-1929), the leading German 
mathematician of the last quarter of the Nine- 
teenth Century. "Mathematics in general is 
fundamentally the science of self-evident things." 


This has been reserved for the last because it is so 
very bad. 

In the first place the modern critical movement 
has taught most mathematicians to be extremely 
suspicious of * 'self -evident things/' In the second 
place it is little better than conceited affectation 
for any mathematician to imply that complicated 
chains of close reasoning are either easy or avoid- 
able from the beginning. After a problem has had 
its back broken by half a dozen virile pioneers it 
is usually simple enough to walk up and dispatch 
the brute with a single well-aimed bullet. If 
mathematics is indeed the science of self-evident 
things, mathematicians are a phenomenally stupid 
lot to waste the tons of good paper they do in 
proving the fact. Mathematics is abstract and 
it is hard, and any assertion that it is simple 
is true only in a severely technical sense that of 
the modern postulational method. The assump- 
tions from which mathematics starts are simple; 
the rest is not. 

Each of the quoted attempts to define mathe- 
matics has contributed a valuable touch to the 
whole picture. These, and the scores of others 
which have not been mentioned, illustrate the 
hopelessness of trying to paint a brilliant sunrise 
in one color. The attempt to compress the free 
spirit of modern mathematics into an inch in a 


dictionary is as futile as trying to squeeze an 
ever-expanding thunder cloud into a pint bottle. 


Less than a century ago it was quite commonly 
thought that mathematics has a peculiar kind of 
truth not shared by other human knowledge. For 
example, Edward Everett in 1870 expressed the 
popular conception of mathematical truth as 
follows : "In the pure mathematics we contemplate 
absolute truths, which existed in the divine mind 
before the morning stars sang together, and which 
will continue to exist there, when the last of their 
radiant host shall have fallen from heaven." 

Although it would be easy to match this extrava- 
gance by many as wild from recent writings of 
those who, like Everett, are not mathematicians 
by profession, it must be stated emphatically that 
only an inordinately stupid or conceited mathe- 
matician would now hold any such inflated esti- 
mate of his trade or of the "truths" he manufac- 
tures. One very modern instance of the same sort 
of thing, and we shall pass on to something more 
profitable. The astronomer and physicist Jfeans 
declared in 1930 that "The Great Architect^ the 
Universe now begins to appear as a pure mathe- 
matician." If this high compliment or that of 


Everett meant anything, pure mathematicians 
might indeed feel proud. 

Against all the senseless rhetoric that has been 
wafted like incense before the high altar of 
"Mathematical Truth," let us put the considered 
verdict of the man whom most professional mathe- 
maticians would agree is the foremost living mem- 
ber of their guild. Mathematics, according to 
Dayid Hiljbert (1862- ), is a game played ac- 
cording to certain simple rules with meaningless 
marks on paper. This is rather a comedown from 
the architecture of the universe, but it is the final 
dry flower of a century of progress. The meaning 
of mathematics has nothing to do with the game, 
and mathematicians pass outside their proper 
domain when they attempt to give the marks 
meanings. Without assenting to this drastic 
deflation of mathematical truth, let us see what 
brought it about. 

The story begins in 1830 with George Peacock 
(1791-1858) and his study of elementary algebra. 
Peacock seems to have been Tme of the first to 
recognize that algebraical formulas are purely 
formal empty of everything but the rules accord- 
ing to which they are combined. The rules in a 
mathematical game may be any that we please, 
provided only that they do not lead to flat contra- 
dictions like "A is equal to B and A is not equal to 


B." The British algebraic school, Peacock, Greg- 
ory (1813-1844), Sir William Rowan Hamilton 
(1805-1865), Augustus De Morgan (1806-1871), 
and others, stripped elementary algebra of its in- 
herited vagueness and embodied it in the strict 
form of a set of postulates. As these postulates 
are illuminating we shall state them in the follow- 
ing chapter in a modern version. Before doing 
so, however, let us see what postulates are. 

A postulate is merely some statement which we 
agree to accept without asking for proof. A 

famous example is Euclid's postulate of parallels, 
one form of which is this: Given a point P in a 
plane and a straight line L not passing through P, 
it is assumed that precisely one straight line L' lying 
in the plane can be drawn through P, such that L 
and L' do not meet however far they are drawn." 
Many geometers after Euclid's time struggled to 
prove that there is one such line I/ and, moreover, 
that there is only one. They failed, for the suffi- 
cient reason that the postulate is incapable of proof. 


We return to this in the next chapter. In passing, 
any modern mathematician will salute Euclid's 
penetrating genius for recognizing that this com- 
plicated statement about parallel lines is indeed a 
postulate, on a level, so far as Euclid was concerned, 
with such a simple postulate as "things which are 
equal to the same thing are equal to one another." 

Euclid's postulate illustrates two points about 
postulates in general. A postulate is not neces- 
sarily "self-evident," nor do we ask "is it true?" 
The postulate is given; it is to be accepted without 
argument, and that is all we can say about the pos- 
tulate itself. In the older books on geometry, 
postulates were sometimes called axioms, and it 
was gratuitously added that "an axiom is a self- 
evident truth" which must have puzzled many 
an intelligent youngster. 

Modern mathematics is concerned with playing 
the game according to the rules; others may inquire 
into the "truth" of mathematical propositions, 
provided they think they know what they mean. 

The rules of the game are extremely simple. 
Once and for all the postulates are laid down. 
These include a statement of all the permissible 
moves of the "elements" or "pieces." 

It is just like chess. The "elements" in chess 
are the thirty two chessmen. The postulates of 
chess are the statements of the moves a player can 


make, and of what is to happen if certain other 
things happen. For example, a bishop can move 
along a diagonal; if one piece is moved to an occu- 
pied square, the other piece must be removed from 
the board, and so on. Only a very original philos- 
opher would dream of asking whether a particular 
game of chess was "true/' The sensible question 
would be, "Was the game played according to the 

Among the permissible moves of the mathe- 
matical game is one which allows us to play. This 
is the assumption outright that the laws of ordi- 
nary logic can be applied to our other postulates. 
As this blanket postulate is of the highest impor- 
tance, we shall illustrate its meaning with a simple 

In the sixth proposition of his first book of Ele- 
ments^ Euclid undertakes to prove that if the 
angles ABC and ACB are equal in the triangle as 
drawn, then the side AB is equal to the side AC. 
His proof is the first recorded example of the in- 
direct method reductio ad absurdum .(reduction to 
the absurd). Euclid provisionally assumes the 
falsity of what he wishes to prove. Namely, he 
assumes that AB and AC are unequal. This leads 
easily to the conclusion that the angles ABC and 
ACB are not equal. But they were given equal. 
Faced with this contradiction, Euclid concludes 


by common logic that his provisional assumption 
that AB and AC are unequal must be wrong. 
Therefore AB and AC must be equal, as this is the 
only way of avoiding the contradiction. 

In this, when fully developed, appeal is made to 
two of the cardinal principles of Aristotelian logic, 
the law of contradjctionjLn^ 


iniddlg. The law of contradiction asserts that no 
A is not-A; the law of the excluded middle asserts 
that everything is either A or not-A. Both of 
these have been accepted until quite recently in all 
sane reasoning, but both, be it observed, are postu- 
lates. As w r e shall see later, the law of the excluded 
middle has been called into question as a univer- 


sally valid part of reasoning within the past 
twenty years by mathematicians. In practically 
all mathematics of the past century, however, the 
whole machinery of common logic has been included 
in the postulates of all mathematical systems. 
Unless otherwise remarked, this assumption is 
tacitly made in everything discussed. 

Having stated a particular set of postulates, say 
those of elementary algebra or those of elementary 
geometry, what next? In the past forty years a 
beautiful art has developed around postulate sys- 
tems as things to be studied for their own sake. 
One question asked about a given set is this. Is 
the set the most economical? Or is it possible to 
prune off one and still have a sufficiency? If so, 
the one that is to be pruned must follow by the 
rules of logic from the others. With a little prac- 
tice even amateurs can construct such desirable 
sets of mutually independent postulates. It is at 
least as amusing a pursuit as solving crossword 
puzzles or playing solitaire, and it is fully as useful 
as whatever anyone cares to mention. 

The requirement of independence for our postu- 
late set is not dictated by necessity but by aes- 
thetics. Art is usually considered to be not of the 
highest quality if the desired object is exhibited in 
the midst of unnecessary lumber. Many an other- 


wise good cathedral has been ruined by too many 

Are the postulates then completely arbitrary? 
They are not, and the one stringent condition they 
must meet has wrecked more than one promising set 
and the whole edifice reared upon it in the past 
hundred years. The postulates must never lead to 
an inconsistency. Otherwise they are worthless. 
If by a rigid application of the laws of logic a set of 
postulates leads to a contradiction, such as *A is 
B and A is not B', the set must either be amended 
so as to avoid this contradiction (and possibly 
others), or it must be thrown away. We shall 
have blundered, and we must start all over again. 
At this point it is pertinent to ask, How do we 
know that a particular set of postulates, say those of 
elementary algebra, will never lead to a contradiction? 

The answer to this disposes once and for all of the 
hoary myth of absolute truth for the conclusions of 
pure mathematics. We do not know, in any single 
instance, that a particular set of postulates is self- 
consistent and that it will never lead to a contradic- 
tion. This may seem strong, but the reader will be 
in a position to judge for himself if he reads the 
succeeding chapters. 

So much for the "absolute truths, which existed 
in the divine mind before the morning stars sang 
together" so far as these were mathematical 


truths, and so much also for the Greater Architect 
of the Universe as a pure mathematician. If he 
he can do no better than some of the postulate 
systems that pure mathematicians have con- 
structed in the past for their successors to riddle 
with inconsistencies, the Universe is in a sorry 
state indeed. The less said about the postulate 
systems for the universe constructed by scientists, 
philosophers and theologians the better. 

If anyone asks where the postulates come from in 
the first place, he is harder to answer. Possibly 
the question is of the kind which mathematicians 
describe as "improperly posed." Merely because 
it sounds like a sensible question is no guarantee 
that it is not as nonsensical as asking when time 




STATEMENT of what 

common algebra is from a modern point of view 
was promised in the preceding chapter. The 
reader is asked to look rather closely at the simple 
postulates given, as from them we shall see pres- 
ently at least one aspect of that process of generali- 
zation which was a distinctive feature of much 
mathematics of the past century. 

The letters a, 6, c in what follows are to be inter- 
preted as mere marks without meaning. Chinese 
characters or f, *, , or any other marks would srf as 
well. The signs , O may be given any names 
we please, for example, tzwgb and bgwzt. For 
the sake of euphony however, they may be read 
plus, times. What follows is a paraphrase of the 
first part of a paper by E. V. Huntington on Defini- 
tions of a Field by Independent Postulates. (Trans- 
actions of the American Mathematical Society, vol. 
4, 1903, pp. 27-37). The whole paper is within 
easy reach of anyone who can read simple formulas. 

The underlying idea is that of what we call a 



class in English. We do not define class, but we do 
assume that, given any class, say C, and an indi- 
vidual, say i, we can recognize intuitively whether 
i is or is not a member of C. If i is a member of C f , 
we say that i is in C. For example, if C is the 
class of horses, and i is a particular cow, we can 
point to i and say i is not in C. All this is so 
simple that the only difficulty is to realize that it is 
less simple than it seems. 

To proceed with common algebra. 

We are given a class and two rules of combination, 
or two operations, that can be performed on any 
couple of things in the class. The operations are 
written , O . We postulate or assume that when- 
ever a and 6 are in the class, the result, written a 
6, of operating with on the couple a, b is a 
unique thing which is in the class. This postulate 
is expressed by saying that the class is closed under 
. We postulate also that the class is closed under 
the operation O. 

A word as to the reading of formulas. Suppose 
a and b are in the given class. By our postulate 
above, a 6 is in the class, and therefore it can 
be combined with any c in the class to give a unique 
thing again in the class. How shall this last be 
written? If we get the result from the couple a 
6, c, we shall write it (a 6) c; if the result is 
got from the couple c, a 6, we shall write it c 


(a 6). At this step the hasty may jump to the 
unjustifiable conclusion that, necessarily, 

(a b) e c = c (a e 6), ' 

where = is the usual sign of equality. 

The only things we shall assume about equality 
are these. 

If a is in the class, then a = a. This says that a 
thing "is equal to" itself. 

If a, 6, c are in the class, and if a = 6 and b = c, 
then a = c. This is Euclid's old friend about 
things equal to the same thing being equal to one 

If a, 6 are in the class, and if a = 6, then 6 = a. 

The postulates proper for common algebra can 
now be stated in short order. In this particular 
set there are seven, which we number for future 

POSTULATE (1 . 1) If a, b are in the class, then a 
b = b a. 

POSTULATE (1 . 2) If a, 6, c are in the class, then 
(a 6) c = a (b c). 

POSTULATE (1 . 3) If a, 6 are in the class, then 
there is an x in the class such that a + x = 6. 

These are merely the familiar properties of alge- 
braic addition precisely and abstractly stated. 
Subtraction is given by (1.3). Notice that our 
covering postulate of closure under permits us 


to talk sense about a b and b ffi a in (1.1), and 
similarly in the rest. The following three make 
common multiplication precise. The postulate 
(2.3) gives algebraic division. 

POSTULATE (2.1) If a, b are in the class, then a 
Ob = b Q a. 

POSTULATE (2 . 2) If a, 6, c are in the class, then 
(a O 6) O c = a Q (6 O c). 

POSTULATE (2.3) If a, b are in the class and are 
such that a a is not equal to a, and b b is not 
equal to 6, then there is a y in the class such that a 
O y = 6. 

The seventh and last connects , O. 

POSTULATE 7. If a, 6, c are in the class, then 
a O (b ffi c) - (a O 6) 8 (a O c). 

Notice that (1 . 1) and (2.1), also (1.2) and (2.2), 
differ only in the occurrence of the signs , O. 

If we now replace by the common +, and O 
by X, and then say that the class shall be that of 
all the numbers, positive, negative, whole or frac- 
tional, that ordinary arithmetic deals with, we see 
that our postulates merely state what every child 
in the seventh grade knows. Of course, to take 
(1.1), (2.1), we must get the same result out of 
6 + 8 as we do out of 8 + 6, and of course 8 X 6 is 
the same number as 6 X 8. 

There is no of course about it. Can it be proved? 
Yes, up to a certain extent, provided we agree to 


stop somewhere and not demand further proof for the 
things asserted. This needs elaboration. 

In common algebra we point to all the numbers 
of common arithmetic, as we did just a moment 
ago, and say there is a class, the numbers, and 
there are two operations, common addition and 
multiplication, which satisfy all our seven postu- 

Examining parts of the curious (2.3), we observe 
that they amount to forbidding the beginners' sin 
of attempting to divide by zero. 

If then we agree to accept common arithmetic as 
a self consistent system, we shall have exhibited a 
consistent interpretation of our seven postulates. 
Otherwise, granted that arithmetic is self-con- 
sistent, we shall have pointed out a self -consistent 
system satisfying our postulates. 

But what about common arithmetic? Why not 
see what it stands on? Do'we know that the rules 
of arithmetic can never lead to a contradiction? 
No. In the past half century a host of mathe- 
maticians have busied themselves over this. Per- 
haps the most striking answer is that which bases 
the numbers on symbolic logic. But on what is 
symbolic logic based? Why stop there? For the 
same reason, possibly, that the Hindoo mytholo- 
gists stopped with a turtle standing on the back of 


an elephant as the last supporter of the universe. 
No finality is possible. 

Another sort of answer was given by Leopold 
Kronecker (1823-1891). An arithmetician by 
taste, Kronecker wished to base all of mathematics 

on the positive whole numbers 1, 2, 3, 4, 

His creed is summed up in the epigram, "God made 
thejntegers, all the rest is the work of man." As 
he said this in an after dinner speech perhaps he 
should not be held to it too strictly. 

In the paper from which the seven postulates are 
transcribed it is proved that the set is independent: 
no one of the seven can be deduced from the other 

The system which the seven postulates define is 
called & field. An instance of a field is therefore the 
common algebra of the schools. The same system, 
namely a field, can also be defined by other sets of 
postulates. There is not a unique set of postu- 
lates for common algebra, but several, all of which 
have the same abstract content. It is just as if 
several men of different nationalities were to 
describe the same scene in their respective lan- 
guages. The scene would be the same no matter 
what language was used. 

Which of all possible equivalent sets of postu- 
lates for a field is the best? The question is not 
mathematical, as it introduces the elements of 


taste, or purpose, or value, none of which has yet 
been given any mathematical meaning. For some 
purposes a set containing the greatest number of 
postulates may be preferable. In such a set most, 
if not all, of the postulates will be simple subject- 
predicate statements. For other purposes a set in 
which not all of the postulates are independent 
might be easier to handle, and so on. 

Before leaving this set, let us recall that it con- 
tains all the rules of the game of common algebra. 
We can make our moves only in accordance with 
these rules. 

We can make any rules we please to begin with 
in mathematics, provided they are consistent. 
But, having made the rules, we must be sportsmen 
enough to abide by them while playing the game. 
If the game should prove too hard or uninterest- 
ing under the prescribed rules, we are free to make 
a new set and play accordingly. The exercise of 
that legitimate license was the source of some of 
the most interesting mathematics of the golden 

We have chosen algebra rather than geometry to 
illustrate postulate systems on account of its 
greater simplicity. The same sort of thing has 
been done repeatedly for elementary geometry, for 
which one of the neatest postulate systems is 
Hilbert's of 1899-1930. 



To recall some useful terms, let us name the rule 
of play given by Postulate (1.1) the commutative 
property of the operation . As the postulate 
(2.1) says exactly the same thing about O that 
(1.1) does about , we refer to it as the commuta- 
tive property of O. Similarly (1.2), (2.2) express 
the associative property, and Postulate 7 is the dis- 
tributive property. These are the familiar names 
of the schoolbooks on algebra. 

The circles in , O can now be dropped, as they 
have sufficiently played their part of emphasizing 
that we are speaking of whatever satisfies the seven 
postulates and nothing else. Accordingly we shall 
now write a + b for a 6, and a . b or ab f or a Q 6, 
exactly as in any text on algebra. 

Suppose now that we rub out one of the postu- 
lates, say (1.2), the associative property for addi- 
tion. Then, whenever a + (b + c) turns up, we 
can not put (a + b) + c for it, as there is no postu- 
late permitting us to do so. We must carry a + 
(b + c) and (a + 6) + c as two distinct pieces of 
baggage, instead of the one piece we had before. 
The new algebra is more complicated than the old. 
Is it any less "true?" Not at all, provided we can 

point to a class of things a, 6, c, and two 

operations, our new "plus" and "times," which 
behave as the six postulates we have now laid down 


require, and which we agree to accept as consis- 
tent. Without bothering for the moment whether 
we can point to an example, let us see how the sys- 
tem defined by the six postulates compares with 
that defined by all seven from which it was derived 
by suppressing one postulate. 

A moment's reflection will show that the new 
system is more general, that is, less restricted, than 
the old. This is plain, because the new system has 
to satisfy fewer conditions than the old, and there- 
fore there is greater freedom within it. Whatever 
we can say about the new system will hold also for 
the old. The other way about is false, for some 
things (namely all those for which the postulate 
(1 . 2) is necessary) can not be said about the new. 

This illustrates one way of generalizing a mathe- 
matical system. We weaken the postulates. 

More than idle curiosity prompts the next ques- 
tion. By weakening the postulates of a field 
(common algebra) how many consistent systems 
can be manufactured? I believe the answer has 
not been printed (it is not mine), but it appears to 
be 1152. At any rate, the mathematicians of the 
past century produced well over 200 such systems 
incidentally in the course of their work on postu- 
lates. There are thus 200 or more, possibly 1151, 
"algebras" in addition to the "common algebra" of 
the schools, and each of these is more general than 


the common one. The schoolboy of the 22nd 
Century may have to learn some of these, but he 
certainly will not be tormented by more than 1152 
in all, for that, it can be proved, is the limit of 
possibilities in this direction. 

Anyone except a mathematician may be par- 
doned for demanding what is the good of this? 
Isn't the algebra of the high school enough for 
practical life? A reasonable answer seems to be 
that high school algebra is either too much or too 
little for everyday life. Only one person in hun- 
dreds ever actually uses the common algebra he 
learned. But for the many in our technical age 
who must use mathematics in their work far more 
than common algebra is desirable and often neces- 
sary. One example must suffice to give some 
weight to this assertion. 

Open any handbook on mechanics or physics as 
they are taught in the first two years of college to 
those who intend to make their livings at applied 
science, and notice the heavy black letters, usually 
in Clarendon type, in the formulas. These repre- 
sent "vectors." A vector is the mathematical 
name for a segment of a straight line which has 
both length and direction. A vector a, inter- 
preted physically, represents, among other things, 
a force of stated amount acting in a stated direc- 
tion. Now follow through a few of the vector 


formulas. Presently the astonishing fact presents 
itself that a X b is not equal to b X a, but to m^nus 
b X a. 

Vectors are added according to our postulates 
(1.1), (1.2); postulate (2.2) is still good, and 
postulate (7) is satisfied, all with perfectly sensible 
physical meanings. But (2.1), the commutative 
property of multiplication, has gone overboard, 
as it is not true for vectors. Instead we have ab = 
-ba. All this, when properly amplified, gives the 
standard vector analysis, without which no one 
would think nowadays of trying to master mechan- 
ics or electricity and magnetism. 

Still stranger specimens of our collection have 
their uses. One, something like vector algebra, 
discovered by W. K. Clifford in 1872, has just 
recently proved of great service in studying the 
complicated mechanics of atoms. Others are of 
equal interest to mathematicians. Even the 
freak we suggested by suppressing (1.2) is not 
without charm. 

Another example of generalization (from geom- 
etry instead of algebra) will be given presently. 
For the moment let us glance back. All that has 
been said is as simple as any interesting game, and 
is in fact far simpler than chess. Its simplicity has 
not bloomed over night. Almost a century was 
required for the perfection of the flower. Sir 


William Rowan Hamilton, ^universal genius and 
one of the most creative mathematicians of the 
golden age, racked his grains for fifteen years in the 
effort to create a suitable algebra for geometry, 
mechanics, and other parts of physics. The 
obstacle which blocked him all those fifteen years 
was the commutative property of multiplication. 
Finally the solution flashed on him one day while 
he was out walking: throw away the commutative 
property; a times b is not always and everywhere 
equal to b times a. 

Today a college freshman discards the commuta- 
tive property without fifteen seconds' thought. 


With a definite system of postulates now at hand 
for inspection, we may ask where they came from. 
To some mathematicians the question is meaning- 
less. Others accept the statement of certain philos- 
ophers that the postulates of mathematical sys- 
tems are derived from experience. This may be 
satisfactory, provided we know what experience 
means. But to say that every set of mathematical 
postulates is a fruit of experience is to stretch the 
meaning of experience to the breaking point, and to 
give an answer that is little better than a quibble. 
If indeed, as Hilbert has asserted, mathematics is 
a meaningless game played with meaningless 


marks on paper, the only mathematical experience 
to which we can refer is the making of marks on 

Instead of trying to answer what may be a sense- 
less question by giving a plausible equivocation 
which any competent mathematician could shoot 
to pieces in two seconds, let us see how one of the 
most celebrated systems of postulates actually 
originated. Anyone who wishes may ascribe the 
postulates already stated for a field to expe- 
rience. The set for Lobachewsky's geometry 
could more properly be credited to a lack of 
experience in any usual sense of the word. 

For centuries before 1826 mathematicians had 
tried to prove Euclid's postulate of parallels (stated 
in the preceding chapter) from the remaining 
postulates of Euclid's geometry. They succeeded 
in proving that if the postulate is so provable, then 
any one of a large number of equivalent geometri- 
cal theorems must be true. Conversely, if one of 
these theorems is a consequence of all of Euclid's 
postulates except the one for parallels, then it can 
be proved that through a point P in a plane can be 
drawn exactly one straight line L' lying in the plane 
determined by P and a straight line L not passing 
through P, such that L and I/ do not meet how- 
ever far extended. 

One of these crucial theorems equivalent to the 


parallel postulate is this 'obvious' trifle. Given a 
segment AB of a straight line, (see figure follow- 
ing) and equal perpendiculars AC, BD erected at 
A and B, and on the same side of AB, join CD, and 
prove that each of the equal angles (they are easily 
proved to be equal) ACD, BDC, marked in the 
figure, is a right angle: 

Common sense at once "sees" that ACD, BDC 
are right angles by folding the rectangle over the 


line perpendicular to AB through the middle point 
M of AB. What common sense thinks it sees is a 
striking illustration of the fact that mathematics 
is not the science of self-evident things. 

Being unable to prove that each of ACD, BDC is 
a right angle by Euclid's geometry without using 
Euclid's parallel postulate, Nicolai Lobachewsky 
(1793-1856) conceived the brilliant and epoch- 
making idea of what is equivalent to postulating 
the assumption that each angle is less than a right 



angle. With minute care he proceeded to develop 
the consequences of this hypothesis. It led him to 
a simple geometry, just as consistent as Euclid's 
and equally sufficient for the needs of everyday life, 
in which he discovered the following undreamed of 
situation regarding "parallels." 

P is any point not on the straight line L; PH is 
perpendicular to L; QT and RS are a particular 



pair of straight lines drawn through P. The 
angle TPS between RS and QT is greater than zero; 
that is, the lines RS arid QT do not coincide. Now, 
in Lobachewsky's geometry, any line I/ passing 
through P and lying within the angle TPS is such 
that it never meets L, however far extended in 
either direction. So then there are an infinity of 
"parallels" in Lobachewsky's geometry. 


In Euclid's geometry, RS and QT coincide, and 
there is only one parallel. Lobachewsky calls the 
two lines PR, PT, neither of which meets L, his 
parallels, as they both have all the properties of 
Euclid's one parallel. 

Which geometry is "true?" The question is 
improper; each is self -consistent. And each is 
sufficient for everyday life. 

But why, out of the three thinkable possibilities 
that the equal angles ACD, BDC in the original 
figure are each less than, equal to, or greater than a 
right angle, stop with the first two, which give the 
respective geometries of Lobachewsky and Euclid? 
There is no compulsion. We may equally well 
postulate that each is greater than a right angle. 
The result is a third geometry, again self-consis- 
tent and sufficient for every day life. In this last 
geometry (developed by Riemann) there are no 
parallels, and a straight line is dosed and of finite 

Why choose Euclid's in preference to either of 
the others? Some would say because Euclid's is 
the simplest of the three to learn, backed as it is by 
2200 years of school teaching. 

The significant thing for us at present is that 
Lobachewsky changed the rules of Euclid's game 
and invented another just as good. This was a 
tremendous step forward. It showed mathemati- 


cians that they might try the same trick of denying 
the obvious, of ignoring or contradicting those 
things which have been accepted in any region of 
mathematics, "always, everywhere, and by all,* 
and see what might come out of their boldness. 

In geometry alone the outcome during the past 
century has been sufficiently staggering. Geome- 
tries by scores have been created and studied inten- 
sively. When first made these were created for 
their own sake. More than once these manufac- 
tured geometries have proved invaluable in science, 
for which the classical geometry of Euclid is today 
quite inadequate. We shall return to this later. 

Before leaving this, however, let us mention 
another direction in which geometry freed itself of 
the shackles of tradition by generalization. Solid 
space for the Greeks had three dimensions, say 
length, breadth and thickness. When geometry 
was studied analytically or algebraically instead of 
synthetically (as was the case up to 1637), the 
restriction to three dimensions no longer was neces- 
sary. It was only in the past century however 
that complete freedom was attained in this direc- 
tion. First, in analytical mechanics in the Eight- 
eenth Century, it became useful to reason about 
solid space and time together as a geometry otfour 
dimensions. The step from/cwr to n (any whole, 
positive, finite number) was taken by Cayley 


(1821-1895). From n dimensions to a countable 
infinity of dimensions came considerably later. 
A countable infinity is as many as there are of all 

the positive whole numbers, 1, 2, 3, From 

geometry of a countable infinity of dimensions to 
an uncountable infinity (as many as there are points 
on a straight line) of dimensions, was the last step, 
taken about thirty years ago. 

If common sense objects to geometry of four 
dimensions, it will get little comfort from modern 
physics. Relativity is based on a particular 
geometry of four dimensions, and geometry of an 
infinity of dimensions is now commonly used in the 
mechanics of atoms. 

The postulational method of setting up mathe- 
matical theories algebra, geometry, and the 
rest was one of the major advances of the past 




o ONE with a musical 
ear would mistake a jig for a waltz. The structure 
of each betrays its nature in the first few bars or 
phrases. Nor would a musician confuse two 
waltzes. Although they belong to the same 
kind of composition, their melodies alone are 
sufficient to distinguish them immediately. 

In mathematics there is frequently discernible 
a similar structure. Within each of several theo- 
ries is an inner harmony of pure form, and the 
form for all is the same. But two theories having 
the same abstract form may be as different in their 
outward appearance and in their applications as 
are two waltzes in sound and emotional appeal. 
This is not intended as more than a rough descrip- 
tion; and the analogy must not be pressed too far. 

As a somewhat crude example, let us look first 
at the postulates of a field stated in the preceding 
chapter. We shall see that common algebra can 
be "realized" in any one of at least three ways. In 
the first the class concerned is that of all rational 
numbers; in the second the class is that of all real 



numbers; in the third the class is that of all com- 
plex numbers. The structure of these three fields 
is the same, namely the postulates (1.1), (1.2), 
(1.3), (2.1), (2.2), (2.3) and (7). Each is, say, 
to pursue the analogy, a waltz; the tunes of all 
three are different. If we work out the conse- 
quences of the postulates once for all abstractly, 
without asking for a tune to lighten our labors, we 
shall have done waltzes completely, all except fit- 
ting melodies to particular waltzes. The melodies 
correspond to the interpretations of the things in 
the given abstract class and those of the abstract 
operations according to which these things are 
combined in accordance with the postulates. We 
use abstract to emphasize that we can say nothing 
about the system considered, here a field, beyond what 
is explicitly stated in the postulates and what can be 
deduced by common logic from those postulates alone. 
When we say, for example, that the things in the 
given class are real numbers, we assert something 
which is not deducible from the postulates, for in 
them the things were mere marks. By thus put- 
ting a definite restriction on the marks, we get a 
field which is no longer abstract or general, but 
special. The formulas for this special field will be 
instances of those for the abstract field. 

Leaving the analogy, we must first describe what 
is meant by rational, real and complex numbers. 


These notions permeate much of mathematics. 
It is assumed that we understand what the zero, 
positive and negative whole numbers, 0, 1, 2, 

, 1, 2 are a vast assumptioh 

in the light of modern critical mathematics. 

If a, b are whole numbers, of which b is not zero, 
the ratio of a to b is a/6 (the result of dividing a 
by 6). A rational number is defined as the ratio of 
two whole numbers. The class of all whole num- 
bers is a subclass of all rational numbers, as is 
seen by restricting the divisor b to be 1. 

The rational numbers do not include the irra- 
tionals. A number is called irrational if it is not 
the ratio of any pair of whole numbers. For 
example, the square root of 2 is irrational, as can 
be easily proved by supposing the contrary and 
getting a contradiction. This fact, by the way, so 
disconcerted Pythagoras, who had constructed his 
theory of the universe on the hypothesis that all 
numbers are rational, that he induced its dis- 
coverer to drown himself in order to suppress the 
awkward theory-destroying fact. So runs the 
story. It is also reported that the fact had become 
so notorious in the golden age of Greece that 
Plato averred that anyone who did not know that 
the square root of two is irrational (he used differ- 
ent words, suited to geometry), was not a man but 
a beast. 


A part of all the irrationals and all the rationals 
are swept up into the common class of real numbers. 
To picture these, take any convenient point on 
an indefinitely extended straight line, and any con- 
venient length, say an inch, which we agree shall 
be the unit of measure. Step off 1, 2, 3, ...... 

inches to the right of 0, and 1, 2, 3, ...... to the 

left; name the first positive, and the second negative. 
The points thus marked, including at O corre- 
spond to the whole numbers. Scattered along the 
line are the points corresponding to the rational 

-* -1 -1 -A. o 

numbers, a few of which are marked in the figure. 
Where is the square root of 2 on the line? To the 
right of O and somewhere between the two rational 
numbers 140/100 and 142/100. Being content for 
the present with that vague somewhere, we remark 
that to each point on the line corresponds one and 
one and only one real number, rational or irrational. 
The real numbers are everywhere dense on the line, 
for between any two we can always locate another 
by bisecting the segment joining the two repre- 
sentative points, if no other way suggests itself. 


The class of all real numbers is the class whose 
members correspond, one-to-one, to all the points 
on the line. 

Complex numbers constitute a still vaster assem- 
blage. In describing them, I shall deliberately 
avoid the perfectly satisfactory way of the high- 
school texts and return to Gauss. This has two 
advantages for our purposes. It avoids the legiti- 
mate but trivial discussion of what "imaginary" 
means. "Imaginary" numbers are no more imag- 
inary than are negatives, if we persist in regarding 
the positive whole numbers as the only true num- 
bers. It also makes it easy to see how mathe- 
maticians in the past seventy years generalized 
complex numbers and invented hypercomplex 

Following Gauss, we let a, b represent any real 
numbers, and we create an ordered couple (a, 6). 
This ordered couple of real numbers is called a 
complex number if it is made to satisfy certain 
postulates, of which we shall state only three as 

The sum (a, b) ffi (c, d) of the given pair of com- 
plex numbers (a, fe), (c, d) is defined to be the com- 
plex number (a + c, 6 + d) ; the result (a, 6) O 
(c, d) of multiplying the given pair (a, fe), (c, d) is 
defined to be (ac-bd, ad + fee) ; equality is defined 
to mean that (a, 6) = (c, d) when and only when 


a = c and b = rf. In the above, a + c, ac, etc., 
have their usual meanings as for real numbers in 

With these definitions of "addition," "multipli- 
cation," and "equality," it is a simple exercise to 



I (+> 


verify that the class of all complex numbers (a, 6), 

(c, d), satisfies all the postulates of a field. 

In passing, we give the usual geometrical picture 
of (a, 6) (fig 6). Through draw a perpendicu- 
lar to the line on which we represented real numbers. 
Take any point P in the plane fixed by these two 
lines, and drop a perpendicular PN to the line of real 


numbers. If the length of ON is measured by the 
real number a, and that of NP when laid along the 
line of real numbers is measured by 6, we affix to 1 
the point P the complex number (a, 6). If P lies 
in either of the quadrants labeled I, IV, a is posi- 
tive; if P lies in II, III, a is negative; if P is in I or 
II, b is positive; if P is III or IV, b is negative. 
The pairs ( + , +), (-, +), (-, -), (+, -), 
taken in the order opposite to the motion of the 
hands of a watch, tell the story on the figure. 
The "imaginary" square root of minus one has not 
been mentioned. Whoever cares to look for it will 
find its image on the vertical line. Notice that 
(a, 6) is also uniquely placed by giving the length of 
OP and the magnitude of the angle NOP, read as 
indicated by the arrow. Now OP is a vector, whose 
magnitude is the length of OP and whose direction 
is NOP. This perhaps suggests why complex 
numbers are of great use in the study of alternating 
currents, where the vectors concerned are repre- 
sented graphically. 

Out of all this several simple and important 
things emerge. First, the infinitely rich class of all 
real numbers is imaged on a mere straight line on 
the plane picturing the class of all complex num- 
bers, which is infinitely rich in straight lines 
they can be drawn in all directions over the whole 
plane. To anticipate a question that will be dis- 


cussed later, we state here one of the great dis- 
coveries of the golden age. Common sense and 
all appearances notwithstanding, there are pre- 
cisely as many real numbers as there are complex 
numbers. Stated geometrically, this says that on 
any straight line in a plane there are just as many 
points as there are in the whole plane. If that is 
not sufficiently jarring to the original sin of our 
preconceived notions, consider this. In the whole 
plane there are only as many points as there are on 
any segment of a straight line, provided only that 
the segment is indeed a segment and has a length 
not zero say a billionth of a billionth of a billionth 
of an inch. There is a still more striking conclu- 
sion of a similar sort. The segment contains as 
many points as there are in the whole of space of a 
countable infinity of dimensions. 

If the reader will look back a few sentences he 
will see the words "one of the great discoveries of 
the golden age." That was not a mere rhetorical 
flourish. It was a historical statement, and was 
meant to be taken literally. It was neither as- 
serted nor implied that a great discovery is ever 
necessarily the final one in a given direction. Out 
of this discovery and what led to it has grown in 
the past twenty years what is today regarded by 
many as the turning point in modern mathematics, 
and we do not yet know whether the signpost reads 
"Go on" or "Go back." 



What else do the rational, the real, and the com- 
plex numbers give us, beyond a nest of Chinese 
boxes each of which is enclosed in the one following 
it? Every schoolboy knows or takes for granted 
that each of the first two classes of numbers satis- 
fies all the demands of common algebra, and those 
slightly more advanced know the same for complex 
numbers. We have then three distinct instances 
of a field three waltzes with different melodies. 
The structure of the field is the same in all three, 
which are abstractly identical; the specialized fields 
differ in their interpretations. 

Before leaving this, we shall answer the natural 
question suggested by the rationals, the reals and 
the complexes. Why not generalize still further, 
say to triples (a, 6, c) of rationals, combined accord- 
ing to appropriate rules? 

The answer is again a great landmark. It was 
proved by Karl Weierstrass (1815-1897) about 
1860, and more simply by Hilbert later, that no 
further generalization in this particular direction 
is possible. We have reached the end of a road. 
As it is of some importance to understand exactly 
what Weierstrass proved, I state it more fully. 
By retaining all the postulates of a field, it is impos- 
sible to construct a -class of things which satisfies all 


postulates and which is not either the class of all com- 
plex numbers or one of the latter 9 s subclasses. 

Here again I have tried to be historically precise. 
It was a landmark of the golden age. In the past 
six years, however, so broad, so rapid and so deep 
is the river of mathematical progress, that this 
landmark has been endangered. Not the fact 
which Weierstrass and Hilbert thought they 
proved has yet been swept away, but the type of 
reasoning which they employed has been called 
into serious question. Professing no opinion on 
these matters, which affect all our reasoning in 
logical patterns inherited from Aristotle, I simply 
report and pass on. 

If complex numbers are the end of this particu- 
lar road, how shall we progress? Go back and 
build another! New roads by hundreds were con- 
structed to higher points of view by the mathe- 
maticians of the century of progress. One great 
highway led to the unbounded field of linear asso- 
ciative algebra, in which the associative property 
of multiplication, but not the commutative, is 
retained in the postulates. 

Having acquired from Lobachewsky, Hamilton, 
and others the habit of denying the obvious, the 
pioneers might easily have contradicted or denied 
one or more of the postulates of a field, as we now 
sometimes do, to reach these vantage points. But 


this is rather a road for the sophisticated, easy 
enough to travel after it has been blasted out of 
the rock and graded. As a matter of fact one of 
the commanding peaks of the Nineteenth Century, 
which we could now reach more easily, was dis- 
covered otherwise and far from naturally. It all 
but revealed itself through the mists a score of 
times to seasoned explorers, who glimpsed its 
lower slopes but never its summit, for almost a 
century before a boy of eighteen looked up and saw 
it all. Less than three years later he was killed in 
a duel. From the summit which Evariste Galois 
(1811-1832) discovered, a host of workers, led by 
Jordan and Kronecker, looked out over the vast 
domain of algebraic equations and algebraic num- 
bers and perceived order, simplicity, and beauty 
in what was chaos to the pioneers; another host, 
led by Felix Klein and ascending yet higher, saw 
the multitude of geometries which the golden age 
discovered as isolated provinces united into a 
single, harmonious pattern of light and shade. 
We shall indicate these summits next. 





ways of looking at old things which seem to be the 
most prolific sources of far-reaching discoveries. 
A particular fact may have been known for cen- 
turies, and it may have been sterile or of only minor 
interest all that time, when suddenly some original 
mind glimpses it from a new angle and perceives 
the gateway to an empire. What the first flash of 
intuition sees may take years or even centuries to 
open up and explore completely, but once a start 
in the right direction is made, discovery goes for- 
ward at an ever increasing speed. Such, in outline, 
appears to have been the evolution of two of the 
dominating concepts of the mathematics of the 
golden age, that of groups and invariants. 

The story begins far back. Distinct traces of 
the long development are discernible in the work of 
the Babylonians and the Greeks who, however, 
never suspected what their regular patterns in 
tilework and other forms of art meant abstractly, 
that is, mathematically. 

A different approach to the dominating idea 



seems to have guided the brilliant Arabian alge- 
braists of the Ninth to the Fifteenth Centuries and 
successive generations of their European followers 
down to the Eighteenth and first two decades of 
the Nineteenth Century. But again those who 
were guided failed to grasp the thread and followed 
it, if at all, subconsciously. 

Regularities and repetitions in patterns suggest 
at once to a modern mathematician the abstract 
groups behind the patterns, and the various trans- 
formations of one problem, not necessarily mathe- 
matical, into another again spell group and raise 
the question what, if anything, in the problems re- 
mains the same, or invariant, under all these trans- 
formations? In technical phrase, what are the 
invariants of the group of transformations? 

When faced with a new problem mathematicians 
frequently try to restate it so that it is equivalent 
to one whose solution is already known. 

In school algebra, for example, the general equa- 
tion of the second degree is solved by "completing 
the square." This reduces the general quadratic 
to one which we can solve at sight. To recall the 
steps: we solve y* = k for y thus, y = db \/k. 
We then reduce ax 2 + 2bx + c = 0, by completing 
the square, to 

/ 6V & 
( x H 1 = 

V / 


which is of the same form as the easy equation y 2 = 
k. In f 

b 2 - ac 

k. In fact, if we now write y x + -, k = 


-, we have exactly y 2 = k. Notice the ex- 

pression ft 2 ac. A remarkable property of this 
simple expression, considered in a moment, started 
the whole vast theory of invariance. 

Successes such as this were some of the reasons 
why mathematicians began to study algebraic 
transformations intensively for their own sake. 
To illustrate a contributory cause, let us consider 
two further simple problems, one from elementary 
algebra, the other from geometry, to see how the 
comprehensive modern concept of invariance 
originated. Those who have forgotten their first 
year of school algebra will have to skip the next. 

In ax 2 + %bxy + cy 2 , express the x, y in terms of 
new letters X, Y as follows, x = pX + qY, 
y = rX + sY. The result is a(pX + qY) 2 + 
2b(pX + qY) (rX + sY) + c(rX + sY) 2 . 

Multiply everything out and collect like terms. 
The result is 

A J 2 + WXY + CY\ 

in which A, B, C are the following expressions in 
terms of a, 6, c, p 9 g, r, s: 


A = ap 2 + Zbpr + c r 2 , 
B = apq -f 

We shall leave it to the reader to verify that the 
new A,B,C and the old a, fe, c are connected by the 
astonishing relation 

B*- AC = (ps - rqY (6 2 - ac). 

To sum up what has happened, let us write 

-f CY\ 
B* - AC = (ps - rqY (& - ac). 

The -4 can be read "is transformed into." The 
indicated transformation of x, y is said to be linear 
(technical term for "of the first degree") in X and 
Y. The expression ps qr, which depends only 
on the coefficients p, q, r, s of the transformation of 
x, y is called the modulus of this transformation. 

Now look at the summary. It says that 6 2 ac 
belonging to the original ax 2 + %bxy + cy 2 , and J5 2 
- AC, belonging to AX 2 + 2BXY + CY\ differ 
only by a factor which is the square of the modulus 
of transformation. For this reason, 6 2 ac is 
called a relative invariant of ax 2 + %bxy + cy* 9 
"relative," because 6 2 ac is not absolutely un- 
changed under the transformation. If however 
p, g, r, s are chosen so that (ps qr} 2 = 1, then 


6 2 ac and J9 2 AC are equal and of the same form, 
and we say that 6 2 ac is an absolute invariant of 
ax 2 + %bxy + cy 2 under the given linear transfor- 
mation. This appears to be the first known in- 
stance of such unchangeableness of algebraic form. 

A mathematician who could look at the relation 
between b 2 ac and B 2 AC and not be at least 
mildly surprised provided it was the first time he 
had seen such a phenomenon would be little more 
than an algebraic imbecile. This elementary fact 
is the acorn, among other things, of the great oak 
which overshadows ipodern physics, Einstein's 
principle of the "covariance of physical laws," and 
it was planted by Gauss in his immortal Disquisi- 
tiones Arithmeticae (published in 1801). Cayley, 
Sylvester, and others made the acorn grow to the 
oak in 1846-1897. 

Our geometrical example requires no algebra. 
Consider the shadows cast on a wall by a book as 
it is turned into various positions. The lengths 
of the sides of the shadow change as the book is 
moved. What does not change? Try it with a 
flat mesh of straight wires. The shadow angles at 
which the wires intersect and the shadow lengths 
of the pieces of wire between intersections change 
in the varying shadows. But an intersection of two 
or more wires remains the same; the shadow wires 


intersect in the same way as the real wires, and the 
straight wires remain straight in shadow. 

The wires represent a simple geometrical con* 
figuration of points (intersections) and straight 
lines. Under the shadow transformation the 
straightness of the lines is invariant. Further, the 
intersection of any number of lines is an invariant 
property, as also is that of the order of any number 
of intersections lying on one straight line. The 
shadow is a particular kind of projection, like that 
of a picture on a screen. 

Let us recall now that the school geometry (Eu- 
clid's) deals almost exclusively with the compari- 
son or measurement of lengths, areas, and angles. 
For instance, the angle inscribed in a semicircle is a 
right angle. What becomes of this under projec- 
tion? It is not invariant, for the circle projects 
into an ellipse and the right angle loses its "right- 


Properties of geometrical configurations which 
are altered by projection are called metric, since they 
depend upon measurements. Properties invariant 
under projection are called protective. This is 
merely a description of terms and not an exact or 
full definition. It is sufficient for our purpose, 
although in passing it may be mentioned that by 
taking account of points whose coordinates are 
complex numbers, the whole of metric geometry can 


be restated more simply as an episode in projec- 
tion. The common non-Euclidean geometries 
also come into the shadow picture. 


Glancing back at the algebraic example and the 
geometrical shadows, we see two general problems, 
one algebraic, the other geometric. 

The geometric one is the more easily stated : Given 
any geometrical configuration, to find all those 
properties of it which are invariant (unchanged) 
under projection. 

This is immediately generalized. Why stop 
with projection, which is only a particular kind of 
transformation? We might for instance seek all 
those properties of extensible, flexible surfaces, 
like sheets of rubber, which are invariant under 
stretching and bending without tearing. The geo- 
metrical problem now is: Given any geometric 
thing configuration, surface, solid, or whatever 
can be defined geometrically and given also a set 
of transformations of that thing or of the space 
containing it, to find all those properties of the 
given thing which are invariant under the trans- 
formations of the set. 

All this can be translated into the perspicuous 
symbolic languages of algebra and analysis. The 
last may be very roughly described as that de- 


partment of mathematics which is concerned with 
continuous variables. A variable is, as its name 
implies, a mark or letter, say x, which takes oui 
different values successively in the course of a given 
investigation. For example, the speed of a falling 
body is not a constant number, say 32 feet per 
second, but a variable whose numerical value in- 
creases continuously from zero (when the body 
starts to fall) to a greatest speed just as the body 
strikes the earth. 

In passing I must apologize for this very crude 
description of variables. To state fully what a 
variable is would take a book. And the outcome 
would be a feeling of discouragement, for our at- 
tempts to understand variables would lead us into 
the present day morass of doubt concerning the 
meanings of the fundamental concepts of mathe- 
matics. I shall ask the reader to trust his feeling 
for language and let it go at that: a variable is 
something which changes. A continuous real 
variable passes through all numbers in a given inter- 
val, say from zero to 10, or from zero to infinity. 

Now, in 1637 Descartes published his epoch mak- 
ing treatise on analytical geometry. At one step 
the whole race of mathematicians strode far ahead 
of the Greek geometers. To understand the con- 
nection between the analytical and algebraical 
aspects of in variance and the geometrical problem 



of invariance, it is essential to see what Descartes 


In the familiar figure below which every school- 
boy (or girl) uses to "graph" one thing or another, 
is the germ of the revolutionary idea. How could 






the whole race have missed this till Descartes saw 
it? A graph is more easily read than any table of 

The point P has the coordinates (x, y} ; x is meas- 
ured along the X-axis XOX', and y is measured 
parallel to the F-axis YOY' '. Distances x to the 
right of are positive, those to the left negative; 


distances y measured up above XOX f are positive, 
those measured down below XOX' are negative. 
The point P is uniquely fixed when its coordinates 
are assigned. 

Now consider all the pairs (x, y) of coordinates 
of a point which satisfy a given equation connect- 
ing x and y ; say the equation is / (x, y) = (read 
as, "equation connecting x and y" or "function 
of x, y equals zero"). All of these pairs will lie on 
a certain line, straight or curved, say briefly on a 
curve, in the plane fixed by XOX' and YOY' and 
/ (x 9 y) = is called the equation of this curve. 
For example, the equation of the circle whose 
centre is at and whose radius is 5 is x 2 + y 2 = 25. 

What Descartes did was this. Instead of study- 
ing curves and surfaces by drawing figures as the 
Greeks had done, he wrote down the equations of 
the curves considered and proceeded to manipulate 
these equations algebraically. Then, conversely, 
he interpreted the resulting algebra in terms of the 
curves whose equations he had written. 

The gain in power was tremendous. A fresh- 
man today can prove with ease properties of curves 
whose difficulties, by the Greek or synthetic 
method, would have taxed the greatest of the 
Greeks. This however does not imply that any 
freshman is necessarily a greater geometer than 
Euclid or Apollonius, or even Pappus. 


The method of Descartes did more. It sug- 
gested literally an infinity of interesting curves and 
surfaces never even imagined by the predecessors 
of Descartes. Many of these are of the highest im- 
portance in practical affairs. 

Another great step forward was rendered possible 
by Descartes' amazingly simple invention of 
analytical geometry. Many prefer pictures, verbal 
or graphic, to equations. The invention of 
analytic geometry enables us to speak the vivid 
language of geometry about things which are alge- 
braic or analytic. 

Last, Descartes potentially freed geometry from 
the unnecessary restriction to space of three dimen- 
sions, although final and complete freedom in this 
respect was achieved only in the present century. 
There is no reason why we should suddenly stop 
with equations in three variables. Why not consider 
any number n? When we do, and use the language 
of geometry, we have a "geometry of n-dimen- 
sional space." This also is of great practical use. 
Thus rigid kinematics is a geometry of 6 dimen- 
sions; the theoretical physics of gases is a geometry 
of 6n dimensions, where n is the number of mole- 
cules in the volume of gas considered. 


A more appropriate question would be "What 
was geometry in its second golden age of the past 


century?" In the last ten or twelve years geom- 
etry has entered a new phase, vaster and more 
powerful than ever. The new geometry goes far 
beyond that which we are about to describe, vast 
as that was, and it is of unprecedented importance 
for its suggestiveness in the physical sciences. 

The spirit of geometry from at least 1872 to 1922 
can not be better or more briefly described than in 
a famous sentence of Felix Klein. All the astound- 
ing inventiveness and infinite variety of geometry 
during that amazingly prolific half century is seen 
as one orderly, simple whole from the commanding 
summit which Klein recognized as the proper point 
of view to sweep in the whole of the past of geo- 
metry and to foresee much of its future. Here is 
the famous sentence: 

"Given a manifold and a group of transformations 
of the same, to develop the theory of invariants relat- 
ing to that group" 

It is a pity to spoil the beautiful simplicity of 
this by explanations, but we can be brief. A 
manifold of n dimensions is a class of objects which 
is such that a particular object in the class is com- 
pletely specified when each of n things is given. 
For instance, a plane is a two-dimensional manifold 
of points, because the plane can be considered as 
the class of all its points and any point in the plane 
is completely specified, or uniquely known, when 


its two coordinates x and y are given. Common 
solid space similarly is a three-dimensional manifold 
of points. I shall leave it to the reader to see that 
common solid space is also a four-dimensional 
manifold of straight lines. This should rob the 
"fourth dimension" of some of its silly mystery. 

The transformations referred to are of the kind 
which replace each object of the manifold by some 
definite object of the manifold, or even of another 
manifold. For instance, we might consider all 
those transformations of the straight lines of solid 
space which carry straight lines into other straight 
lines, or into spheres, for (as the reader may easily 
think out for himself) common solid space is four- 
dimensional in spheres as well as in lines. (It 
takes four numbers to fix a particular sphere; 
three to fix the coordinates of the centre, and one to 
fix the length of the radius) . The number of dimen- 
sions of any space depends only upon the elements 
(points, lines, planes, spheres, circles, etc.} in terms 
of which the space is described. 

The transformations, according to Klein, must 
form a group. The postulates for a group are 
given in the next chapter, and these postulates are 
the official definition of a group. But as the group 
is the central and commanding concept of Klein's 
whole vast program, let us describe its leading 


Consider a class of things and a set of opera- 
tions which can be performed on the members of 
that class. If the result of performing any one of 
the operations upon any given member (or mem- 
bers) of the class is again a member of the class, we 
say that the class has the group property with 
respect to the operations. The class then is closed 
under the operations of the set. Thus the class of 

positive whole numbers 1, 2, 3, has the 

group property with respect to addition, for the 
sum of any two of these numbers is again one of 
the class. The like holds also for multiplication, 
but not for subtraction or division. 

The invariants in Klein's program are those 
things (properties, actual figures, or what not) 
that persist, or remain unchanged, under all the 
transformations, or operations, of a particular 
given group. 

Finally, notice that nothing is said about the 
number of dimensions of the manifold. This may 

be 1, or 2, or 3, or n, or it may be infinite. 

All possibilities are envisaged in the vast program. 

Was Klein's program simply an empty dream, 
an unnecessary abstraction and generalization of 
the familiar? Far from it. From that single 
point of view the geometers of the golden age saw 
projective geometry, metric geometry of all kinds, 
Euclid's geometry, innumerable non-Euclidean 


geometries, geometries of any number of dimen- 
sions, and much more, as harmonious parts of 
Klein's comprehensive, simple program. It was 
one of the memorable things of all mathematical 
history, not merely an outstanding achievement of 
the past century. That the present is going be- 
yond Klein, and ascending higher than he saw, 
does not diminish the sublimity of his conception. 



Jr ROM Klein's program 

for geometry it is clear that the concept of a group 
dominated at least one major province of mathe- 
matics during the past century. Groups also were 
found to be the structure behind much of modern 
algebra, in particular the theory of algebraic equa- 
tions. Wherever groups disclosed themselves, or 
could be introduced, simplicity and harmony 
crystallized out of comparative chaos. Finally 
some modern philosophers became interested in 
this powerful, unifying mathematical concept of 
groups as an important phase of scientific thought. 
As the idea of a group was one of the outstanding 
additions to the apparatus of scientific thought of 
the last century, we shall discuss it at some length. 
Before proceeding to the official definition of an 
abstract group, I add a word of caution. Vast as 
was the panorama swept in from the vantage point 
of groups, it by no means included the whole of 
mathematics, either ancient or modern. In many 
a fertile mathematical province groups either play 
no part or only a very subordinate one. The 




whole theory of groups itself is but an incident in 
the algebra of the past century. 

Groups are first subdivided into two grand divi- 
sions, finite and infinite. The number of distinct 
operations in a finite group is finite; in an infinite 
group the number of distinct operations is infinite. 
The subject was developed in the Nineteenth Cen- 
tury by a host of mathematicians, among whom 
Galois, Cauchy, Jordan, Lie and Sylow may be 

A finite group according to a famous dictum of 
Cay ley in 1854 is defined by its multiplication 
table. Such a table states completely the laws 
according to which the operations of the group are 
combined. Here is a specimen which can be easily 

/ A B c D E 




I A B C D E 

A B I D E C 

B I A E C D 

C E D I B A 

D C E A I B 

E D C B A 1 


This group contains the six operations I, A, B, 
C, D, E. We shall state what the table says about 
any pair of these operations, say B and D. Take 
any letter, say B, from the lefthand vertical column, 
and any letter, say D, from the top horizontal row, 
and see the entry C in the table where the B-row 
and the D-column intersect. It is just as if we 
were to multiply B by D, say B X D, and get the 
answer C. Instead of writing B X D, we shall 
write BD, which says to take B from the left, D 
from the top, and find where the corresponding row 
and column intersect. This gives the result C; so 
we write BD - C. 

What about DB, found according to the same 
rule? It is not equal to C, but to E; namely, DB 
= E. So in this kind of composition, BD and DB 
are not necessarily equal. The reader may easily 
satisfy himself that although the commutative 
law has gone, the associative is still valid. For 
instance (AB) C = A (BC). 

Let now x be any member of a given class on 
which I, A, B, C, D, E operate. We postulate that 
the result of operating with any one of /, A, B, C, D, 
E on x gives another member of the class. Let us 
write B (x) (read, "J? on or") for the result of operat- 
ing on x with B. By our postulate this is some 
member of the given class, so we can operate on 
B (x) with D. The result is written BD (x), which 


is again in the class. Now, the assertion of the 
table that BD = C says that, instead of performing 
the operations J5, D successively, first B and then 

D, we could reach the same final result in one step, 
by performing the operation C on x. Thus, the 
class is closed with respect to the operations I, A, B, 
C, D, E. For the results of performing the opera- 
tions of the set successively are always in the set. 
If the reader does not believe this, let him follow 
the rule which gives BD = C, DB = E, CE = A, 
EC = jB, etc., and try to escape from the table. 
Lay aside this book for a moment and reflect on 
the miracle that such closed, finite sets actually 

Notice the effect of operating with /. The table 
says that AI = I A = A; BI = IB = J?, and so for 
all. Thus / as an operation changes nothing; it is 
called the identity. 

The last thing to be observed attentively is this. 
Given any one of /, A 9 B, C, D, E, say X, there is 
always exactly one other of the six, say Y such that 
XY = I. Further, for every such pair, X, Y it is 
true that XY = YX. It is not asserted that X, Y 
are necessarily distinct. For example, if X is the 
particular operation B, then the table says that 
Y = A, because BA = 7; if X is E, then Y also is 

E. Two operations X, Y such that XY = the 
identity are called inverses of one another. The 


table states that each element of the set has a unique 

A set of operations having all of the foregoing; 
properties is called a group. The definition by 
postulates will be given presently. 

For the moment let us see that an instance of the 
group defined by the specimen multiplication table 
actually exists. There are dozens of them all in 
different parts of mathematics. Here is a very 
simple specimen from arithmetic. Start with any 
number different from zero, say x. We can sub- 
tract xfrom 1, and we can divide 1 by x 9 getting the 
new numbers 1 x and I/a:. Repeat these opera- 
tions on the new numbers. Then 1 x gives back 
x and a new number I/ (1 x);l/x gives the new 
number 1 I/a; or (x 1) fx, and gives back x. 
Keep this up forever. You can never get but one 
or other of the six numbers x, I/a;, 1 x, I/ (1 x), 
(x !)/#, x/ (x 1). Now let 7 be the operation 
which transforms x into itself, / (x) = x; let B be 
the operation which transforms x into (x 1) /x, 
or B (x) = (a; l)/av and so on, with C (x) = 
I/a:, D (x) = 1 - x, E (x) = x/(x - 1). A little 
patience will show that these I, A, B, C, D, E 
satisfy the multiplication table. 

The number of different operations in a group is 
called its order. Thus our group is of order 6. 
Looking at the table more closely, we see several 



smaller groups within the whole group, for example 
those whose multiplication tables are 

/ C 

I A B 

1 \ I , I 

1C. I 

I A B > 


C 1 A 

A B I 


B I A 

whose respective orders are 1, 2, 3. Now 1, 2, 3 
are divisors of 6, and we have illustrated a funda- 
mental theorem of groups, the order of any sub- 
group of a given group is a divisor of the order of the 

The following postulates for a group should now 
be intelligible. 

We consider a class and a rule, written as o, by 
which the two things A, B in any ordered couple 
(A, B) of things in the class can be combined so as 
to yield a unique thing which is again in the class. 

The result of combining A, B in the ordered 
couple (A, B) where A and B are any things in the 
class, is written A o B. 

POSTULATE. (Closure under o). If A, B are in 
the class, then A o B is in the class. 


POSTULATE. (Associativity of o). If A, B, C 
are in the class, then (A o B) o C = A o (B C). 

POSTULATE. (Inclusion of identity) . There is fc 
unique thing I in the class such that A o / = I o A 
for every thing A in the class. 

POSTULATE. (Unique inverse). If ^4 is any 
thing in the class, there is a unique thing, say A', 
in the class such that A o A' = /. 

The foregoing postulates define a group: the class 
is said to be a group under (or with respect to) the 
composition o. The postulates contain redundan- 
cies, but are more easily seen in the above inelegant 

form. The A, B, C, are our previous 


It is instructive to compare the postulates for a 
group with those for a field. It will be seen that, 
if we suppress the commutative property of multipli- 
cation in a field, the remaining postulates for 
multiplication are those of a group, and likewise 
for addition. 

If the composition o does have the commutative 
property (as in the arithmetical examples above), 
the group is called commutative, or Abelian (after 

Let us glance back here at the linear transforma- 

*= P X + qY\ 


of Chapter V. Merely write down the coefficients 
p, g, r 9 s of this transformation, as Cayley did in a 
fundamental "Memoir on Matrices" in 1858, thus 

p. q 

r, s 

This is called a matrix (of order 2, since there are 2 
rows and two columns) . What of it? the skeptical 
reader may ask. I refer him to the physicists for 
the moment, until an answer can be indicated 
shortly. At present I wish merely to emphasize 
that matrices were invented in 1858 by Cayley for 
the purposes of pure mathematics, and neither he 
nor anyone else dreamed that 88 years later they 
would prove to be a subtle clue to some of the 
deepest mysteries of the physical universe. 

Cayley dealt directly with the matrix instead of 
with the linear transformation of which it is the 
skeleton. An important thing about matrices is 
the way they are combined, or operated upon, or, in 
technical phrase, multiplied. The rule is illus- 
trated thus 

r, s 


pP + qR, pQ + qS 
rP -f sR t rQ + sS 

where X is read "times," and the matrix on the 
right of the sign = is by definition the result of 



performing X on the given matrices in the given 
order. As numerical examples: 

1, 3 

2, 4 

3, 5 
6, 7 



3, 5 

6, 7 

1, 3 

2, 4 

21, 26 

30, 38 

13, 29 

20, 46 

The multiplication of matrices is not commutative, 
as shown by this example or the rule. 

Suppose now that we perform two linear trans- 
formations in succession. The matrix of the single 
resulting linear transformation is obtained by the 
rule above. The extension to matrices of any 
order is immediate. From this remark it may be 
surmized that groups and matrices are intimately 
connected, and this is the fact. Cayley and his 
successors perfected the theory of matrices; the 
theory of groups is a mine for our successors to 

Let us glance back. No man, I believe, no mat- 
ter how practical, could point to a more conspicu- 
ous example than matrices of the apparently futile 
things over which mathematicians labor as few 
others ever dream of laboring. And it would be 
difficult to find a better instance of the historical 
fact that the significant advances of mathematics 


and not a few of those of science are inspired by the 
spirit of pure mathematics. Just as "beauty is its 
own excuse for being," so mathematics needs no 
apology for existing. I apologize, on the contrary, 
for pointing out presently that matrices do have a 
non-mathematical use, and a highly important one 
at that. This use, by the way, is but one of many. 

After such an instance as this, scientists, educa- 
tors, and others may be more willing to let mathe- 
matics develop according to its own nature, in- 
stead of insisting, as they have sometimes done, 
that it should draw its life from finance, bridge- 
building, statistics, or whatever happens to be 
popular at the moment in physics. As remarked 
before, the deliberate effort to follow immediate 
utility in mathematics almost invariably leads to 
second or third rate work, and more often than 
not the very utility which is narrowly sought turns 
out to be not so great after all. 

The mathematics of what many mean by every- 
day life, practical mechanics, buying and selling, 
and the other necessary activities by which we live 
more or less from hand to mouth, is for the most 
part worked out and reduced to simple rules of 
thumb. But that which is of vital importance in 
modern life, which is based on an ever expanding 
science and an ever more scientific technology, is 
not simple, and it is not a matter for the engineer- 


ing handbooks. It is partly in process of creation. 
Mathematicians may safely be left to follow their 
own bent as their contribution to this age of science. 
What they did in the past century is enough for a 
vast region of science and technology as they exist 
today; what mathematicians as professionals are 
interested in today will, if there is any continuity 
at all in scientific and industrial history, be the 
indispensable framework of the science and tech- 
nology of tomorrow. 

To return to matrices, one of the fair examples of 
the foresight of mathematics. When Heisenberg 
in 1926 was casting about for some adequate means 
of formulating the mechanics of the atom pos- 
sibly the dominant interest in Twentieth Century 
physics he found exactly what was required in 
the theory of matrices, with its queer "multiplica- 
tion" in which a times b is not necessarily b times 
a. From that work developed with amazing speed 
the new mechanics and the new physics of spectra 
and atoms. 

In 1926, and again in 1931, Hermann Weyl 
wrought the new physics into a beautiful, sugges- 
tive pattern in which the theme is the theory of 
groups and the interpretation of quantum mechan- 
ics in terms of that great, abstract theory. In 
this interpretation some of the most abstract and 
advanced labors of algebraists for the past 70 years 


are drawn on and pressed into service to the limit. 
The advanced analysis, integral equations, and 
the rest, of the past thirty years is also utilized to 
the full. 


A word must be said about infinite groups. 
These again fall into two grand divisions. In the 
first, the distinct things are denumerable, that is, 
the things in the group can be counted off 1, 2, 3, 

, but we never come to the end. Such groups 

are infinite and discrete. In continuous groups 
the number of distinct things is infinite, but not 
denumerable; the things can not be counted off 1, 

2,3, , but are as numerous as the points on 

a line. 

Continuous groups arise in the following way 
among others. In school geometry it is assumed 
that a plane figure, say a triangle, can be moved 
about all over the plane and retain its shape (size 
of angles and length of sides) . Consider the group 
of all motions of a rigid figure in a plane. Evi- 
dently the group contains infinitesimal transfor- 
mations, for we can shift the figure from one posi- 
tion to another by stages as small as we please. 

Another example of a group consisting of infini- 
tesimal transformations is that of the rotations of 
a rigid, solid body about a fixed axis. Either the 


body as a whole may be thought of as being moved 
from one position to another, or the motion may be 
realized by subjecting each individual point to an 
appropriate transformation. Both points of view 
are useful. 

Now let us recall that the equations of mechanics 
and those of classical mathematical physics are 
differential equations. Roughly, such equations 
express laws concerning rates of change of one 
or more continuously varying magnitudes with 
respect to one or more others. As a simple ex- 
ample, the velocity of a falling body is the "rate" 
of change of position with respect to time. The 
vast theory of differential equations was greatly 
furthered by the introduction of continuous groups 
into its study. For instance, the central equa- 
tions of higher dynamics, those named after their 
discoverer, Hamilton, when viewed from the stand- 
point of continuous groups become much clearer 
than before. 

The study of such groups absorbed the working 
lives of many mathematicians from 1873 to the 
early years of this century, when interest dimin- 
ished, owing to a great memoir published in 1894 by 
Elie Cartan, which disposed of several of the main 
problems. The theory of continuous groups in its 
broad aspects was almost exclusively the work of 
one man, Sophus Lie (pronounced Lee), 1842- 


1809. But with the new physics, beginning with 
general relativity in 1915, and continuing with the 
quantum mechanics of 1926- (?), continuous 
groups suddenly were seen to be of fundamental 
importance in the description of nature. New and 
yet more general geometries, suggested partly by 
physics, are being created in swarms, and in this 
outburst the theory of continuous groups has been 
at least a highly suggestive guide. In this latest 
renaissance of infinitesimal geometry, Cartan has 
been and is a leader. 

The foregoing is mentioned to introduce another 
landmark of progress. In the past seven years 
Klein's great program, which directed geometry 
for half a century, has been found insufficient. 
Geometry is blooming again, more freely and more 
luxuriantly than ever, uncramped by the limita- 
tions imposed by the theory of groups. The new 
geometries of the highest suggestiveness for 
physical science do not conform to the pattern of 
the group. Galileo was right. The world does 


Although it is not our intention to discuss special 
results, we may close this description of groups by 
referring to one which would have delighted Py- 
thagoras and have caused him to sacrifice at least a 


thousand oxen to his immortal gods. The story 
covers nearly 2200 years. Only the high points 
can be indicated. 

The Greek geometers early discovered the 5 
regular solids of Euclidean space, the tetra- 
hedron, cube, octohedron, dodecahedron, and ico- 
sahedron of 4, 6, 8, 12 and 20 sides respectively, and 
proved that there are no others. This discovery 
begot much of the incredible mysticism of later and 
less exact thinkers. 

Our next high point is about 2000 years farther 
on. For over two centuries algebraists had tried 
in vain to solve the general equation of the fifth 

a* 8 + bx* -f ex 3 + dx* + ex + /= 0, 

until Abel in 1826 and Galois in 1831 proved that it 
is impossible to express x by any combination of 
the given numbers a, 6, c, d, e,f, using only a finite 
number of additions, multiplications, subtractions, 
divisions and extractions of roots. Thus it is 
impossible to solve the general equation of the fifth 
degree algebraically. On the eve of that stupid 
duel in which he was killed, Galois, then in his 21st 
year, wrote out his mathematical testament, 
in which, among other tremendous things, he 
sketched a great theorem concerning all algebraic 
equations. He reduced the problem of the alge- 


braic solution of equations to an equivalent, ap- 
proachable one in groups. As this is an outstand- 
ing landmark in algebra, I shall state Galois' 
theorem, in the hope that some may be induced to 
go farther and find out for themselves exactly what 
it means: an algebraic equation is algebraically 
solvable, if, and only if, its group is solvable. No 
more technical knowledge is necessary to follow 
the proof than is possessed by high school gradu- 
ates. As a consequence of this perfect theorem, 
it is impossible to solve the general equation of any 
degree greater than 4 algebraically. 

In 1858 Hermite solved the general equation of the 
fifth degree, not algebraically, for that would have 
been to do the impossible, which is too much even 
for mathematicians, but by expressing x in terms of 
elliptic modular functions (a sort of higher species 
of the familiar trigonometric functions). 

Our last peak was discovered by Klein, who 
showed in 1884 that the profound work of Hermite 
was all implicit in the properties of the group of 
rotations about axes of symmetry which change an 
icosahedron into itself that is, which twirl the 
solid about so that, say, a given vertex slips over 
to the place where some other vertex was, and so 
for all in every rotation. There are 60 such rota- 

That the rotations of an icosahedron and the 


general equation of the fifth degree should be 
unified from the higher standpoint of groups, is a 
good illustration of the power of the concept of an 
abstract group 

The far reaching power of the theory of groups 
resides in its revelation of identity behind apparent 
dissimilarity. Two theories built on the same 
group are structurally identical. The more famil- 
iar is worked out; the results are then interpreted 
in terms of the less familiar. 



FAUSS crowned arith- 
metic the queen of mathematics. Gauss lived 
from 1777 to 1855, and to his profound inventive- 
ness is due more than one strong river of mathe- 
matical progress during the past century. He 
also made outstanding contributions to the science 
of his time, notably to electromagnetism and 
astronomy. His opinions therefore carry weight 
with all mathematicians and with some scientists. 

Arithmetic to Gauss, as to the Greeks, was pri- 
marily the study of the properties of whole num- 
bers. The Greeks, it may be remembered, used a 
different word for calculation and its applications 
to trade. For this practical kind of arithmetic the 
aristocratic, slave-owning Greeks seem to have had 
a sort of contempt. They called it logistica, a 
name which survives in the logistics of one modern 
school in the logic and foundations of mathematics. 

In arithmetic as in all fields of mathematics du- 
ring the past century discovery went wide and far. 
But there was one most significant difference be- 
tween this advance and the others. Geometry, 



analysis, and algebra each acquired one or more 
vantage points from which to survey its whole 
domain; arithmetic did not. 

The Greeks left no problem in geometry which 
the moderns have failed to dispose of. Faced by 
some of the trifles which the Greeks left in arith- 
metic we are still baffled. For instance, give a rule 
for finding all those numbers which, like 6, are the 
sums of all their divisors less than themselves, 
6 = 1+2+3, and prove or disprove that no odd 
number has this property. To say that arithme- 
tic is mistress of its own domain when it cannot 
subdue a childish thing like this is undeserved 

The theory of numbers is the last great uncivil- 
ized continent of mathematics. It is split up into 
innumerable countries, fertile enough in them- 
selves, but all more or less indifferent to one an- 
other's welfare and without a vestige of a central, 
intelligent government. If any young Alexander 
is weeping for a new world to conquer, it lies before 

Lest this estimate seem unduly pessimistic, let 
us not forget that in each of the several countries 
of arithmetic there was considerable progress in 
the past century. Indeed, two or three of the 
splendid things done are comparable to anything 
in geometry, with this qualification, however: no 


one advance affected the whole course of develop- 
ment. This possibly is due to the very nature of 
the subject. 

Among the notable advances is that which 
revealed one source of some of those mysterious 
harmonies which Gauss admired in the properties 
of whole numbers. This was the creation by 
Kummer, Dedekind and Kronecker of the theory 
of algebraic numbers. In this particular field the 
invention of ideal numbers is comparable to that 
of non-Euclidean geometry. Another striking 
advance was the brilliant development of the 
analytic theory of numbers during the past thirty 
years. Of isolated problems inherited from the 
past that have been successfully grappled with we 
may mention in particular Waring's of the Eight- 
eenth Century. Another result of singular inter- 
est was the proof that certain numbers are tran- 
scendental, and the construction of many such 
numbers. We shall briefly indicate the nature of 
all these things presently. These preliminaries 
may well be closed with the following quotation. 

"The higher arithmetic/' wrote Gauss in 1849, 
"presents us with an inexhaustible storehouse of 
interesting truths of truths, too, which are not 
isolated, but stand in the closest relation to one 
another, and between which, with each successive 
advance of the science, we continuously discover 


new and wholly unexpected points of contact. A 
great part of the theories of arithmetic derive an 
additional charm from the peculiarity that we 
easily arrive by induction at important proposi- 
tions, which have the stamp of simplicity upon 
them, but the demonstration of which lies so deep 
as not to be discovered until after many fruitless 
efforts; and even then it is obtained by some 
tedious and artificial process, while the simpler 
methods of proof long remain hidden from us." 


The positive, zero and negative whole numbers of 
common arithmetic are called rational integers, to 
distinguish them from algebraic integers, which are 
defined as follows, 

Let a , ai, a 2 , , a n _i, a n be n + 1 given 

rational integers, of which a is not zero, and not 
all of which have a common divisor greater than 1. 
It is known from the fundamental theorem of alge- 
bra (first proved in 1799 by Gauss) that the equa- 

a x n + fli x"' 1 + . . . -f a-i x + a n 

has exactly n roots. That is, there are exactly n 

real or complex numbers, say Xi, # 2 > x n9 

such that if any one of these be put for x in the 
equation, the left hand side becomes zero. Notice 


that no kind of number beyond the complex has to 
be created to solve the equation. If n = 2, we 
have the familiar fact that a quadratic equation 
has precisely two roots. For emphasis I repeat 

that a , i, a 2 , , a n in the present discussion 

are rational integers, and that a is not zero. The 
n roots #i, # 2 , > #n are called algebraic num- 
bers. If a is 1, these algebraic numbers are called 
algebraic integers, which are a generalization of the 
rational integers. For instance, the two roots of 
Sx 2 + 5x + 7 = are algebraic numbers; the two 
roots of x 2 + 5x + 7 = are algebraic integers. 

A rational integer, say n, is also an algebraic 
integer, for it is the root of x n = 0, and so 
satisfies the general definition. But an algebraic 
integer is not necessarily rational. For instance, 
neither of the roots of x 2 + % + 5 = is a rational 
number, although both are algebraic integers. In 
the study of algebraic numbers and integers we 
have another instance of the tendency to generali- 
zation which distinguishes modern mathematics. 

Omitting technical details and refinements, we 
shall give some idea of a radical distinction between 
rational integers and those algebraic integers which 
are not rational. First we must state what a field 
of algebraic numbers is. 

If the left hand side of the given equation ao# n + 
a L x n ~ l + + a n = 0, in which a , ai, , 


a n are rational integers, can not be split into two 
factors each of which has again rational integers 
as coefficients, the equation is called irreducible 
of degree n. 

Now consider all the expressions which can be 
made by starting with a particular root of an irre- 
ducible equation of degree n (as above) and operat- 
ing on that root by addition, multiplication, sub- 
traction and division (division by zero excluded). 
Say the root chosen is r; as specimens of the results 
we get r + r, or 2r, r/r or 1, r X r or r 2 , then 2r 2 , 
and so on indefinitely. The set of all such expres- 
sions is evidently a field, according to our previous 
definitions; it is called the algebraic number field of 
degree n generated by r. This field will contain 
algebraic numbers and algebraic integers. It is 
these integers at which we must look, after a slight 
digression on rational integers. 

The rational primes are 2, 3, 5, 7, 11, 13,17,19, 

23, 29, , namely the numbers greater than 1 

which have only 1 and themselves as divisors. 
The fundamental theorem of arithmetic states that a 
rational integer greater than 1 is either a prime or 
can be built up by multiplying primes in essen- 
tially one way only. For instance, 100 = 2 X 2 X 
5 X 5, 105 = 3 X 5 X 7. This is so well known 
that some writers of school books assert it to be 
"self-evident," which is another instance of the 


danger of the obviqus_in mathematics^ Euclid 
gave a beautiful proof of this theorem, one of the 
gems of all his work. If the reader has never seen 
a proof, it may puzzle him to make one. 

Primes in algebraic numbers are defined exactly 
as in common arithmetic. But the "self-evident" 
theorem that every integer in an algebraic number 
field can be built up in essentially one way only by 
multiplying primes is, unfortunately, false. The 
foundation has vanished and the whole superstruc- 
ture has gone to smash. 

One should not feel unduly humble at jumping 
to this particular obvious but wrong conclusion. 
More than one first rank mathematician less than 
a century ago did the same. One of them was 
Cauchy, but he soon pulled himself up short. In 
some algebraic number fields an algebraic integer 
can be built up in more than one way by mul- 
tiplying primes together. This is chaos, and the 
way back to order demanded high genius for its 

The way in which the whole question originated 
was this. Fermat (1601-1665) bequeathed this 
teaser to his exasperated successors. "It is impos- 
sible to find three rational integers x, y, z all differ- 
ent from zero, and a rational integer n greater than 
2, such that x n + y n = z n . The exception 2 is 
necessary; for instance, 3 2 + 4 2 = 5 2 , 5 2 + 12 2 = 


The mathematician and business man Fermat 
was well known for his probity. He asserted that 
he had an "extremely simple proof," which the 
margin of his book was too narrow to contain. 
For nearly 300 years arithmeticians and others 
have broken their heads over Fermat's assertion 
and so far haven't made a dent in it. The asser- 
tion remains unproved, although it is known to be 
true for numerous values of n. A general proof is 
what is wanted. 

About 1845 E. E. Kummer (1810-1893) thought 
he had one. His friend Dirichlet pointed out the 
mistake. Kummer had assumed the truth of that 
obvious but not always true theorem about the 
prime factors of algebraic integers. He set to 
work to restore order to the chaos in which arith- 
metic found itself, and in 1847 published his resto- 
ration of the fundamental law of arithmetic for the 
particular fields connected with Fermat's assertion. 
This achievement is usually rated as of greater 
mathematical importance than would be a proof of 
Fermat's theorem. To restore unique factoriza- 
tion into primes in his fields, Kummer created a 
totally new species of number, which he called 

In 1871 Richard Dedekind (1831-1916) did the 
like by a simpler method which is applicable to the 
integers of any algebraic number field. Rational 


arithmetic was thereby truly generalized, for the 
rational integers are the algebraic integers in the 
field generated by 1 (according to our previous 
definitions) . 

Dedekind's "ideals," which replace numbers, 
stand out as one of the memorable landmarks of 
the past century. I can recall no instance in 
mathematics where such intense penetration was 
necessary to see the underlying, true pattern be- 
neath the apparent complexity and chaos of the 
facts, and where the thing seen was of such shining 
simplicity. A rough idea of Dedekind's "ideals" 
can be glimpsed from their very degenerate form 
for the rational integers. 

Consider the fact that 3 divides 12 arithmeti- 
cally. The quotient is 4. Therefore 12 is four 
times as great as three. The last is precisely what 
we must not look at, obvious and true as it is. On 
the contrary we need to see that 4, from another 
point of view, is really bigger than 12, in the sense 
of greater inclusiveness. Precisely: we no longer 
look at 3 and 12, but at the respective classes of 
rational integers which we get when we multiply 
each of 3, 12 by all the rational integers in turn. 
Thus, some of the integers in the class so generated 
by 3 are -9, -6, -3, 0, 3, 6, 9, , and simi- 
larly from 12 we have the specimens 36, -24, 
-12, 0, 12, 24, 36, The class generated 


by 3 is called the principal ideal 3, and similarly for 
12. The ideal 3 contains, or includes the ideal 12. 
That is, every rational integer in the ideal 12 is in 
the ideal 3. The other way about is false; for in- 
stance, the ideal 12 does not contain 9. 

The reader may easily see that all the properties 
of common arithmetical division persist if we make 
the following changes: replace every rational 
integer by the principal ideal which it generates, 
and replace the word "divides" by "contains." 

It was a natural extension of this inverted way 
of looking at division which restored the funda- 
mental theorem of rational arithmetic to the vaster 
domain of algebraic number fields. The extension 
deals with classes defined, not by a single integer, 
but by a set of n integers. 

As might be expected from the definition of 
algebraic numbers as roots of algebraic equations, 
the theory of groups plays an important part in 
algebraic number fields. Here also the marvellous 
creations of Galois in the theory of algebraic equa- 
tions have free scope. Exploration in this terri- 
tory is still in progress, and much is being dis- 


A mere glance at these must suffice. A number 
which is not algebraic is called transcendental. 
Otherwise stated, a transcendental number satis- 


fies no algebraic equation whose coefficients are 
rational numbers. It was only in 1844 that the 
existence of transcendentals was proved by Joseph 
Liouville (1809-1882). The transcendental num- 
bers, hard as they are to find individually, are 
infinitely more numerous than the algebraic num- 
bers. A very famous transcendental is w (pi), 
the ratio of the circumference of a circle to its 

diameter. To 7 decimals, <* = 3.1415926 , 

and it has been somewhat uselessly computed to 
over 700. In 1882 Lindemann, using a method 
devised in 1873 by Hermite, proved that TT is tran- 
scendental, thus destroying for ever the last slim 
hope of those who would square the circle al- 
though many of them don't know even yet that the 
ancient Hebrew value 3 of T was knocked from 
under them centuries ago. 

In 1900 Hilbert emphasized what was then an out- 
standing problem, to prove or disprove that 2^ 
is transcendental. The rapidity of modern prog- 
ress can be judged from the fact that Kusmin in 
1930 proved a whole infinity of numbers, one of 
which is Hilbert's, to be transcendental. The 
proof is quite simple. 


Fermat proved that every rational integer is a 
sum otfour rational integer squares. Thus 10 


O 2 + O 2 + I 2 + 3 2 , 293 - 2 2 + 8 2 + 9 2 + 12 2 , etc. 
In 1770 E. Waring guessed that every rational 
integer is the sum of a fixed number N of n th 
powers of rational integers, where n is any given 
integer and N depends only upon n. For n = 3, 
the required N is 9; for n = 4, it is known that N 
is not greater than 21. Hilbert, by most ingenious 
reasoning, proved Waring's conjeccture to be cor- 
rect in 1909. In 1919 G. H. Hardy (1878- ) 
applying the powerful machinery of modern analy- 
sis, gave a deeper proof, the spirit of which is 
applicable to many other extremely difficult ques- 
tions in arithmetic. This advance was highly 
significant for its joining of two widely separated 
fields of mathematics, analysis, which deals with 
the uncountable, or continuous, and arithmetic, 
which deals with the countable, or discrete. 

Finally, quite recently, Winogradov has brought 
some of these difficult matters within the scope of 
comparatively elementary methods. Here again 
progress is increasingly rapid. The conquests 
being made today in this field would have seemed to 
the men of a hundred years ago to be centuries 
beyond them. 


In our generation we have seen the application 
of analysis to arithmetic on a scale which only 


fifty years ago was undreamed of. As one type of 
problem in this province, we may cite the distribu- 
tion of primes. The question is to state how many 
primes there are below a given limit, say a billion 
billion. To find and count them is humanly impos- 
sible. The problem as stated seems to be hope- 
less; an exact, terminated formula in terms of 
simple expressions is out of the question. But we 
can ask for a formula of this kind: if P (x) denotes 
the number of primes not greater than x, to find an 
expression containing x such that P (x) divided 
by this expression tends to the limiting value 1 as 
x tends to infinity. This has been solved; the re- 
quired expression is x divided by the logarithm of x. 
The solution of this age-old problem was given in 
1896 by J. Hadamard and de la Vallee Poussin 
simultaneously and independently. Subsequent 
work deals, roughly, with estimating the error com- 
mitted in distribution formulas by stopping short 
of the end. A little more precisely, the analytic 
theory of numbers is largely concerned with deter- 
mining the order (relative size) of the errors made 
if we take an approximate enumeration in a par- 
ticular problem concerning a class of numbers in- 
stead of the exact enumeration. In this the 
leaders are G. H. Hardy, Edmund Landau, and J. 
E. Littlewood. 

The broader significance of all this work is its 


fusion of modern analysis and arithmetic into a 
a powerful method of research in the theory of 
numbers. Fifty years ago such a union would 
have been an idle dream. 

One further problem may close this sketch. In 
1742 Chr. Goldbach, on only scanty empirical 
evidence, stated that every even positive rational 
integer is a sum of two primes, for example 30 = 
13 + 17. All the available data seem to substan- 
tiate this wild guess in the dark. It has success- 
fully resisted all analytical (and other) attacks. 
But the mere fact that modern analysis can take 
hold of and handle a problem of such inhuman 
difficulty is an indication of progress. 



1 NFINITY and the infinite 

have long had a singular fascination for human 
thought. Theology, philosophy, mathematics and 
science have all at some stage of their development 
succumbed to the lure of the unending, the un- 
countable, the unbounded. "Only infinite mind 
can comprehend the infinite," according to one; 
"Cantor's doctrine of the mathematical infinite is 
the only genuine mathematics since the Greeks," 
according to another, while yet a third, contradict- 
ing both, declares that "the infinite is self incon- 
sistent, and Cantor's theory of the mathematical 
infinite is untenable." 

Here we reach a frontier of knowledge, and fur- 
ther progress will necessarily be slow. Some be- 
lieve that mathematics is about to retrace many of 
the giant strides it made toward the infinite in the 
past half century; others foresee a steady progress 
in the direction already travelled. 

The simple fact seems to be that no one at pres- 
ent can say exactly where mathematics stands 
with regard to its supposed conquest of the infi- 



nite, and no one can sensibly predict its future. 
Equally competent authorities hold diametrically 
opposing views. 

With this caution against accepting anything in 
what follows as final, we may proceed to a short 
description of the kind of scaling ladders with 
which mathematicians "stormed the heavens," in 
Weyl's phrase, during the past fifty or seventy 


The infinite entered mathematics early. Not to 

go too far back, let us glance at one type of prob- 
lem which induced Archimedes in the Third Cen- 
tury B.C. to use the infinite in mathematics. It 
is that of finding the area under a curve, say the 
area ABCD. Cut the area up into strips of equal 
breadth, and disregard the shaded triangular bits. 
The remaining rectangles can easily be calculated. 


Their sum is an approximation to ABCD. If by 
taking thinner and thinner rectangles the sum of 
the discarded shaded bits tends to zero, and if the 
sum of the rectangles tends to a limit, this limit is 
the required area. To reach the limit we must take 
the sum of an infinity of rectangles. This crude 
description must suffice. 

With the invention of the calculus in the Seven- 
teenth Century and its applications to the finding 
of areas, surfaces, and volumes of all imaginable 
shapes, such infinite summations, known as inte- 
grations, became one of the most powerful tech- 
niques of analysis. 

Mathematical physics could not exist without 
integration. Consider for example the simple 
problem of calculating the work done as a variable 
force moves a body through a given distance, work 
being measured as force times distance in the 
proper units. 

The process inverse to integration or summation 
is called differentiation. It will be sufficient here 
to state one geometrical application of differentia- 
tion. To draw a tangent line at a given point of a 
given curve necessitates the finding of the slope of 
the tangent line, and this is equivalent to perform- 
ing a specific differentiation. Now consider this. 
It is intuitively evident that we can always draw a 
tangent to a continuous curve at a given point of 


the curve. Intuition is misleading; there exist 
continuous curves which have no tangents at all. We 
admit gladly that this is shocking to common, 
sense, for it shocked mathematicians when Weier-J 
strass first confronted them with such a curve in 

Now let us go back to summation a moment. 
The solutions of multitudes of mathematical and 
physical problems lead to infinite sums. Here are 
three specimens: 

1 - i + i - i+ ...... ; 

1 + x* + * + z* + ..... ; 

These are almost pathologically simple, but they 
will do. The dots mean that the series are to con- 
tinue without end, according to the law indicated in 
each case. Now, it can be proved that the first 
series converges to a definite, finite number as we 
proceed to infinity, adding and subtracting the 
f r actions as they occur. If x is a real number, the 
second series converges only for such x as lie be- 
tween 1 and +1 ; for all other real values of x the 
series diverges, that is, by adding a sufficient num- 
ber of terms, the sum can be made to surpass any 
previously assigned number. The third series is 
divergent, although it does not look it. It seems 
incredible that the sum of a sufficient number of 


terms of this series can be made bigger than a 
billion billion billion, but such is the fact. An- 
other astonishing thing is that the first series is not 
equal to (1 + J + | - + ...... ) - (i + i + i + 

Is it not clear that if a physical problem, say the 
calculation of a temperature, yields as answer a 
divergent series, then that answer has no physical 
meaning? When such nonsense turns up we go 
back, revise our mathematics and reformulate the 
problem, or give it up. 

One of the outstanding things young Abel and 
Cauchy did in the early decades of the Nineteenth 
Century was to provide the first methods whereby 
the convergence of a series can be tested. 

From the foregoing handful of examples we can 
appreciate the program of that great triumvirate 
Weierstrass, Dedekind, and Cantor, who in 1859 
to 1897 undertook a thorough examination of the 
mathematical infinite itself. Another impulse to 
an attack on the infinite was the problem of irra- 
tionals. What does the square root of two mean, 
if it is not the ratio of any pair of whole numbers? 

Dedekind's attack on irrationals is a modern 
reverbration of Eudoxus. If either falls under the 
counter attack of modern skeptics, both fall. 
Paradoxical as it may seem, the last conclusion is 
no novelty of the Twentieth Century. Isaac 


Barrow, the teacher of Newton, late in the Seven- 
teenth Century acutely criticized Eudoxus, and 
Barrow's objections to the logic of the great Greel 
have been repeated by the leading Twentieth 
Century critics of the mathematical theory of the 
infinite elaborated by Weierstrass, Dedekind, and 
Cantor. If nobody listened to Barrow, the like 
cannot be said for Brouwer and his school. 

Let us look at one or two of the central concepts 
of this controversial subject. Mathematical anal- 
ysisthe calculus and every luxuriant growth that 
has sprung from its fertile soil in the past two cen- 
turies derives its meaning and its life from the 
mathematical infinite. Without a firm founda- 
tion in the infinite, mathematical analysis treads 
at every step on dangerous ground. 


Let us consider first what counting means. At 
a glance we see that the two sets of letters x, y, z 
and X , F, Z contain the same number of letters, 
namely 3. We say that two classes contain the 
same number of things if the things in both classes 
can be placed in one-to-one correspondence, that is, 
if we can pair of the things in the two classes and 
have none left over in either. For example, we 
can pair x with X 9 y with F, z with Z. We say 
that two classes are similar if the things in them 
can be paired in one-to-one correspondence. 


Observe this simple fact : the classes x, y, z, w and 
X, Y, Z are not similar. Try as we will, we cannot 
find a mate for some one of x, y> z, w. The reason 
here is plain; the first class contains/owr things, the 
second only three, and four is greater than three. 
Everyone saw this for thousands and thousands of 
years and, for a wonder, everyone saw straight. 
The next took genius of a high order to perceive. 
Georg Cantor (1845-1918) is the hero of this. 

Consider all the positive rational integers 

1, 2, 3, 4, 5, 6, 7, 8, .... 

and under each write its double, thus, 

1, 2, 3, 4, 5, 6, 7, 8, ... 

2, 4, 6, 8, 10, 12, 14, 16 

How many numbers 2,4,6, are there in the 

second row? Exactly as many as there are num- 
bers altogether in the first, for we got the second 
row by doubling the numbers in the first. The 

class of all the natural numbers 1, 2, 3, 4, is 

similar to a part of itself, namely to the class of all 

the even numbers 2, 4, 6, 8, There are just 

as many even numbers as there are whole numbers 

This illustrates a fundamental distinction be- 
tween finite and infinite classes. An infinite class 
is similar to a part of itself; a finite class is similar to 


no part of itself. "Part" there means proper part, 
namely, some but not all. 

As another example, let us see that any two seg- 
ments of a straight line contain the same number 
of points. (For brevity I am forced to omit many 
refinements which a mathematician would de- 
mand, but the following illustrates what is meant.) 
Suppose the segments AB and CD are of different 

lengths. Place them parallel as in the figure, and 
let AC, ED meet in 0. Take any point, say Q, 
on AB, and join OQ. Let OQ cut CD in P. This 
sort of construction puts the class of all points on 
AB into one-to-one correspondence with the class 
of all points on CD. 

Is there no escape? What about postulating that 
the points on a line are not dense everywhere, but 
strung like dewdrops on a spiderweb, and that 
any segment contains only a finite number of 


points? Such finite, discrete geometries have been 
extensively investigated by American mathema- 
ticians in the past thirty years by the postula- 
tional method. But to say that space whatever 
scientists and others mean by space is gran- 
ular in structure and not continuous is too repug- 
nant to habit to be acceptable. Nevertheless, in 
physics, energy parted lightly enough with some of 
its continuity in 1900 when Planck quantized it, 
to avoid mathematical and physical absurdities. 
Instead of quantizing space, mathematicians at 
present prefer to overhaul their reasoning. 


As analysis rests on numbers, I interpolate here 
an answer to the question of what a cardinal num- 
ber is, say 2, or 3, or 4, or any other number which 
states "how many." The answer was given in 
1884 by G. Frege, whose work passed almost un- 
noticed, possibly because much of it was written 
in an astounding symbolism which looked as com- 
plicated as a cross between a Babylonian cuneiform 
inscription and a Chinese classic in the original. 
It is the finest example of the precept that mathe- 
maticians should write so that he who runs may 
read. Bertrand Russell independently arrived at 
the same definition in 1901, and expressed it in 
plain English. Here it is : 


The number of a class is the class of all those classes 
that are similar to it. 

This is not meant to be simple. It is profound, 
and it is worth pondering until one grasps its truth 
intuitively. Beside this gem of abstract thought 
the visions of the mystics seem material and gross. 


How did Dedekind tame the irrationals? We 
postulated that the square root of 2 can be repre- 
sented by a point on the line of all real numbers, 
lying somewhere between 1 and 2, and that, by 
approximating more and more closely we can 
narrow the interval in which the elusive number 
lies. But to trap it alone, and not get a whole 
brood of undesirables in the trap at the same time, 
requires supreme skill. 

Dedekind provided this in his famous "cuts," 
which can be applied at any point of the line of 
reals. We need consider only that kind of cut 
which separates all the rational numbers into two 
classes of the following sort : each class contains at 
least one number: every number in the "upper" 
class is greater than every number in the "lower" 
class. Further, the numbers of the upper class 
have no least number, those of the lower class have 
no greatest number. 

We can now imagine the "upper" and the 


"lower" classes laid down on the line of real num- 
bers. Owing to those provisos about no greatest 
and no least in the respective classes, the two 
classes will, as it were, strive to join one another. 
But they cannot, because any number in the upper 
is greater than every number in the lower. The 
place where they strive to join is the cut, and it 
defines some irrational number. 

To locate the square root of 2 as a cut, we put 
into the upper class all those positive rational num- 
bers whose squares are greater than 2, and into 
the lower class all other rational numbers. A 
moment's visualization will reveal that the elusive 
square root of 2 is definitely trapped between the 
two classes and is in the trap alone. 

The Dedekind cut is at the root of modern mathe- 
matical analysis. Another root of that ever fer- 
tile tree is the vast theory of assemblages which, 
roughly, discusses among other things the proper- 
ties of curves, surfaces, and so on, as sets or classes 
of points. An outstanding problem in the theory of 
sets is this: Can the elements in any set whatever 
be well ordered? For example, consider all the 
points on a segment of a straight line. Between 
any two points of the line we can always find 
another point of the line. How then shall we 
individualize this uncountable infinity of points 
and call each by its name according to any conceiv- 


able system of nomenclature? We do not know. 
A very famous postulate, Zermelo's of 1904, prac- 
tically assumes that any assemblage can be \\ell 
ordered, for his unaccepted proof rests on a doubt- 
ful postulate. The postulate asserts that if we are 
given any set of classes, each of which contains at 
least one thing, and no two of which have a thing 
in common, then there exists a class which has just 
one thing in each of the classes of the set. Why 
should this be true, if it is, of an infinite set of 
classes? This assumption, like all of the notions 
sketched in this chapter, has been challenged. It 
is much less innocent than it looks. We may have 
reached the great turning point in the progress of 
mathematics, and we may have to retrace our 
steps or swerve to one side to circumvent the 
unsurmountable. Whatever happens, we shall 
have lived through an epic age. 

In the final chapter we shall indicate further 



WAS remarked of 
Africa, there is always something new coming out 
of analysis. This vast domain comprises every- 
thing that concerns continuously varying quanti- 
ties. Its importance for natural science is there- 
fore evident, since it is true, apparently, that "all 
things flow." Fixity is an illusion, and analysis 
gives us a firm grasp on the laws of continuous 

The progress in analysis during the past century 
was beyond all precedent. Today its scope is so 
vast that probably no mathematician is competent 
in more than a province or two of the entire domain, 
Particularly is this so if, as seems legitimate, we 
include under analysis the modern developments of 
differential geometry the investigation of geo- 
metrical curves, surfaces, and so on, from the 
study of configurations and structure in a small 
neighborhood. The last man to look out over the 
whole field of analysis was the universally-minded 
|3enri Poincare (1854-1912), and he was able to 
do so largely because great tracts of modern analy- 


sis were his own creations. On practically every 
department of mathematics this outstanding 
genius left his deep impression. 

In all of this bewildering progress it is not easy 
to find commanding points of view from which to 
survey any significant expanse of the whole, un- 
bounded territory. The boundaries in all direc- 
tions are being pushed forward so rapidly that the 
eye soon loses them in the distance. 

Nevertheless the past hundred years did indicate 
one or two general directions of advance, at which 
we must look. It may be said that three of the 
leading activities were the invention and exploita- 
tion of new species of functions in almost incon- 
ceivable variety, continual generalization, and 
drastic criticism of the foundations on which analy- 
sis rests. 

Standards of rigor in proof were constantly 
raised. What had passed as satisfactory at an 
earlier period was minutely scrutinized, often 
found to be shaky, and firmly established according 
to the standards of the day. In this direction 
finality is not sought, for it is apparently unattain- 
able. All that we can say is, in the words of a 
leading analyst, "sufficient unto the day is the 
rigor thereof." 

Another tendency manifested itself. No sooner 
was a significant advance made in another depart- 


ment of mathematics than analysis seized upon the 
central ideas and assimilated them with voracious 
speed. Thus groups, invariants, much of geom- 
etry and parts of the higher arithmetic succes- 
sively become its more or less willing prey. On the 
other hand, wherever it was found possible to 
apply the techniques of analysis to any other 
domain, whether purely mathematical or scientific, 
the advance was swift and sure. 


Nowhere more strongly than in analysis do we 
appreciate the peculiar power of mathematical 
reasoning. This power is traceable, at least partly, 
to the fact that mathematics does not direct iso- 
lated or individual weapons at a problem, but 
unites whole complexes of subtle and penetrating 
chains of thought into new, intimately wrought 
engines of reason, often expressed by a single 
symbol whose laws of operation are once for all 
investigated, and then applies these as units to 
the problem on hand. It is somewhat like the 
advance of an entire, well coordinated army by a 
single order, instead of fussing over the details by 
which the individual companies are to manoeuvre. 
The mere creation of the single weapon begets 
unsuspected power in the parts of which it is com- 
posed and, operating as a unit, the whole achieves 


incomparably more than the sum of the achieve- 
ments of the parts. Unsuspected possibilities 
present themselves automatically. Before the 
designer of the new weapon is aware of it, he has 
made a conquest of which he never dreamed. 

Instance after instance of this peculiar power 
might be cited. We shall sketch only two briefly. 
In each it was not mathematics alone which won 
the victory. The insight, or intuition, of a great 
physicist was in each case necessary before the 
physical problem could be formulated mathe- 
matically. But neither advance could have been 
made certainly neither was made without 
powerful mathematical analysis. The ability to 
translate new scientific problems into mathe- 
matical symbols appears to be as rare as the genius 
which creates the mathematics to solve the prob- 

Our first example goes back to 1864. In that 
year James Clerk Maxwell (1831-1879), having 
translated some of Michael Faraday's brilliant 
experimental discoveries in electromagnetism into 
a set of differential equations, and having filled out 
the set of equations to fit a physical hypothesis of 
his own, proceeded to manipulate the equations 
according to standard processes of mathematical 

Now, one of the fundamental equations in mathe- 


matical physics expresses the fact that whatever 
satisfies the equation in a given instance is propa- 
gated throughout space in the form of waves. 
Moreover the equation contains the velocity with 
which the waves are propagated. 

Manipulating his electromagnetic equations, 
Clerk Maxwell derived from them the wave equa- 
tion of mathematical physics. The indicated ve- 
locity was that of light. Whether he was surprised 
at what the mathematics gave him, he does not 
record. At any rate he proceeded to exploit his 
discovery in grand fashion. He showed that 
electromagnetic disturbances must be propagated 
through space as waves. Further, from the man- 
ner in which the velocity entered the equation, he 
concluded that light is an electromagnetic disturb- 

This was in 1864. Clerk Maxwell died in 1879. 
In 1888, Heinrich Hertz (1857-1894), directly in- 
spired by Clerk Maxwell's prediction of "wireless" 
electromagnetic waves, and guided by his prede- 
cessor's mathematics, set out to produce the waves 
experimentally and to determine their velocity. 
From his success has sprung the whole wireless and 
radio industry of today, and it all goes back to a 
few pages of mathematical analysis. But again 
we must emphasize that without Clerk Maxwell's 
extraordinary skill in setting up the equations and 


his physical intuition, the mathematics could not 
have got very far. On the other hand Hertz might 
never have even started. 

Our second example goes back to 1854, when 
Bernhard Riemann (1826-1866) lectured before 
the venerable Gauss, "Prince of Mathematicians," 
"On the Hypotheses which lie at the Foundations 
of Geometry." This work pleased Gauss greatly, 
as it was a worthy sequel to his own of many years 
previously. One of Riemann's ideas concerned 
measurement in a curved space of any finite num- 
ber of dimensions. In a curved two-dimensional 
space, for example the surface of a sphere, the 
formula of Pythagoras for the square on the 
longest side of a rightangled triangle, on which all 
everyday measurements of distance are based, does 
not hold. Riemann supplied a perfectly general 
formula, good for any of an infinity of "spaces" in 
which the curvature changes in any sufficiently 
general manner from point to point. Near the 
conclusion of his remarkable dissertation he made 
the following striking prediction of one great ad- 
vance of the Twentieth Century. 

"Either therefore the reality which underlies 
space must form a discrete manifold, or we must 
seek the ground of its metric relations outside it, 
in binding forces which act upon it. 

"The answer to these questions can only be got 


by starting from the conception of phenomena 
which has hitherto been justified by experience, 
and which Newton assumed as a foundation, and 
by making in this conception the successive 
changes required by facts which it cannot explain." 

He goes on to say that narrow views and preju- 
dice must not hamper the free investigation of all 
the novelties he has suggested. 

Riemann's new geometry was one of the tower- 
ing landmarks of the past century. A host of 
workers developed it, including E. B. Christoffel 
'(1829-190Q), after whom the famous "index sym- 
bols/' familiar to physicists through relativity, are 
.named. In all this work the theory of invariants 
played a leading part. 

In the 1880's the geometer Ricci started a new 
development in Riemannian geometry. This was 
tied up with the concept of invariance. Ricci 
developed a calculus of extraordinary power for dis- 
covering those geometrical properties in Rieman- 
nian space which are invariant under extremely 
general (in fact almost any) transformations. This 
calculus is called now tensor analysis. 

Consider now the statement of any physical fact 
or "law." If this statement contains essential ref- 
erences to the observer's particular way of expres- 
sing the law, then the supposed law is as much an 
expression of his tastes as of nature's. The point 


need not be labored. Einstein saw it (nobody else 
had till he pointed it out in 1915 as one of the cor- 
nerstones of his general relativity), and today it is 
appreciated by all who honor Einstein. 

While Einstein was constructing his general 
theory in 1906 to 1915, he cast about for some cal- 
culus which would yield the differential equations of 
mathematical physics in invariant form. Covari- 
ant is the usual technical term, but in this connec- 
tion it means the same. He found what he wanted 
in the calculus of Ricci. Einstein also needed an 
adequate geometry to describe the four-dimen- 
sional physical world of space-time. He found it 
in the work of Riemann and his successors. The 
rest was physics plus supreme genius, and is so 
well known today that it need not be repeated. 

When general relativity first came out physicists 
were appalled at the unfamiliar mathematics the 
commonplaces of over half a century to profes- 
sional mathematicians. Today serious students of 
physics take all this in their stride and think no 
more of Christoffel symbols than they do of any 
other necessary mathematical tool. At the lead- 
ing French technical school, PEcole Polytech- 
nique, tensor analysis is taught along with mechan- 
ics in the second year of the regular course. It is 
far more practical than the older vector analysis. 

Relativity has generously repaid its debts to 


geometry and analysis by starting a new golden 
age in geometry. 


A distinctive feature of progress in analysis dur- 
ing the past century was the stupendous develop- 
ment of the theory of functions of a complex vari- 
able. We saw earlier that if all the postulates of 
common algebra are retained, then no numbers 
more general than the complex satisfy the postu- 
lates. This gives a strong hint why functions of a 
complex variable sweep up so much of analysis. 
The development had well started by 1830; in 
fact Cauchy had then made his greatest contribu- 
tions to this branch of analysis, of which he was 
the creator. 

In the period we are considering two other ways 
of looking at the whole subject were discovered, 
one by Weierstrass, the other by Riemann. 

Weierstrass arithmetized analysis. His univer- 
'sal tool was the power series. He regarded func- 
tions from the point of view of the convergent 

infinite series (like a + a\z + a 2 z 2 + + 

a n z n + ) which define them for values of the 

variable z in ranges appropriate to the functions. 

Riemann on the other hand may be said to have 
geometrized the analysis of functions of a complex 
variable. By a most ingenious model of connected 


sheets or surfaces superimposed on a plane, for in- 
stance, he gave an intuitive picture of the proper- 
ties of certain highly important complex functions, 
particularly those which take several different 
values for a given value of the variable. This 
development contributed greatly to analysis situs, 
the geometry which studies the properties of sur- 
faces, volumes and the like which are invariant 
under a continuous group of transformations. 

To give any adequate idea of this vast field in a 
few words is impossible, and we must pass on. 
Touching analysis situs however, we may mention 
one unsolved elementary problem which can be 
stated untechnically. 

Practical map makers have never found a map, 
no matter how complicated, which cannot be col- 
ored, as a map should, by four colors. Prove that 
four colors are sufficient for any map in which con- 
tiguous countries have a line boundary in common. 
This looks easy. The solution may turn out to be 
extremely simple. The problem has important 
bearings on several others. 


Many of the most interesting functions which 
were intensively investigated from 1830 to 1900 
were discovered from 1800 to 1830. Here again 
the field is too vast for more than a slight glance. 


We shall mention only one type of functions which 
claimed the attention of analysts from about 1880 

Consider first any periodic phenomenon say 
the passage of the tip of the minute hand of a 
watch over the 12-o'clock mark. This passage is 
made at regular intervals of one hour. We say 
that the position of the tip is a periodic function of 
the tune, with period one hour. Periodic phenom- 
ena permeate science. Wave motion is an in- 
stance. For this reason, if no other, periodic func- 
tions were extensively investigated by the analysts 
of the past century and a half. 

Expressing the periodicity in the above example 
algebraically, we write/ (t + 1) = / (f), which is 
read, "function of / + 1 is equal to function of t." 
Here "function" may be interpreted as position 
expressed in terms of time t. The thing to be 
noticed is that the numerical value of / (t) is unal- 
tered when we replace the variable t by the linear 
expression t + 1, depending only upon t. Thus the 
value of the function is invariant under a particular 
linear transformation of the variable. 

Why stop here? Poincare in the 1880's went 
much farther, and considered functions invariant 
under groups (in the technical sense explained pre- 
viously) of linear transformations of their variables. 
The result was a new kingdom of analysis. 


As a byproduct of all this, Poincare solved the 
general algebraic equation of the nth degree by 
showing how its n roots can be expressed explicitly 
in terms of some of the functions he had created. 

Finally, throughout the whole century, func- 
tions with any finite number of periods received 
much attention. These alone supply enough for 
a life's work. 


What is perhaps the most striking generaliza- 
tion originated in the work of Vito Volterra 
(1862- ) and his school in the 1880's and 90's. 
In a word, Volterra investigated functions of a 
non-denumerable infinity of variables ,- a non-den um- 
erable infinity being as many as there are of points 
on a straight line. For example, instead of looking 
at a curve as a relation between the coordinates of 
any point on it, we may consider the curve itself as 
the variable thing, and see what happens as one 
curve shades into another. The curve however, 
from another angle, is the set of all its points, and 
this set is non-denumerably infinite. 

This, and what grew out of it, appears to be the 
true mathematical approach to all those physical 
problems where all the past history of a given 
thing has to be taken account of in predicting the 
future. For example, a steel bar when magnetized 


and then demagnetized, exhibits more or less per- 
manent modifications which must be included in 
the mathematical analysis. Here and elsewhere, 
in economics for instance, the theory of integral 
equations and its modern extensions, largely the 
work of the past thirty years, is the clue. The 
subject originated with Abel and Murphy (who 
was a clergyman) . 

Roughly the distinction between such equations 
and those of the classical physics is this. In the 
classical mechanics and physics it is rates of change 
which enter the equations (differential equations) ; 
in the newer work it is the inverses of such rates, or 
integrations (infinite summations) which appear. 
From a given relation between these it is required 
to disentangle the functions which are integrated. 
Finally, in 1906, Henri Lebesgue revolutionized 
integration itself. 

Those in a position to judge predict that these 
comparatively new fields will presently prove to be 
of an importance in science at least equal to that 
of differential equations, which have dominated 
physical science for over two centuries. 

In differential equations the expansion during 
the past eighty years also has been enormous. 
Some of this was inspired directly by physics, much 
of it not. 


We alluded in the first chapter to boundary 
value problems. Such a problem is of the follow- 
ing type. Suppose we know the equation (as we 
do) which expresses the law of conduction of elec- 
tricity in a medium, say in a sheet of copper of any 
shape. Applying the current at any parts of the 
sheet, we wish to know the subsequent distribution 
of electricity over the whole sheet. This is accom- 
plished by making the general solution of the known 
equation satisfy the initial physical conditions. It 
is clear from the physics of the situation that once 
these conditions are given, say the places where 
the current is supplied and the amounts supplied 
there, the solution is uniquely determined. There 
cannot be conflicting distributions at any place at 
any time. Fitting of solutions of equations to 
prescribed initial conditions is technically known 
as solving boundary value problems. 

This again leads to a vast field, still under vigor- 
ous development, in which many of the special 
functions devised by analysts in the past find their 
scientific interpretations. 



T. .. 
o THE uninitiated it 

may seem a very queer proceeding to build up vast 
systems of knowledge without seeing first whether 
the foundations will bear the superstructure. 
Mathematics did precisely that. As weaknesses 
began to appear in the foundations, and one part 
or another of the colossal edifice crumbled, mathe- 
maticians made hasty repairs and went on build- 
ing, until more serious faults made themselves 
evident, and so on for well over a century. 

Who shall criticize the builders? Certainly not 
those who have stood idly by without lifting a 

There is nothing reprehensible in the way mathe- 
maticians have worked. Any creative artist 
knows that criticism before a work is fairly com- 
plete is ruinou. . Only after the work is far enough 
along to be offered to the public is criticism rele- 
vant when it cannot cause the artist to spoil his 

The critics of mathematics have been mathe- 
maticians almost without exception. The one 



reputable exception is Bishop Berkeley who, in 
the Eighteenth Century, showed that he knew 
what he was talking about when he gave the New- 
tonians a run for their money. In general the 
matters in dispute lie far below the surface, and 
are not likely to be observed by any but mathe- 
maticians as they go about their business. 

In passing, let us record that Berkeley's specific 
criticisms were not met until the second half of the 
Nineteenth Century, when Weierstrass drove out 
of analysis the "infinitesimals," or "infinitely small 
quantities," of the Newtonians, to which Berkeley 
had so vigorously objected. 

An anecdote concerning the arithmetically- 
minded Kronecker foreshadows one phase of the 
modern objections to mathematical reasoning. 
When everyone was congratulating Lindemann in 
1882 over his proof that TT is transcendental (see 
chapter VII), Kronecker said, "of what value is 
your beautiful proof, since irrational numbers do 
not exist?" Here Kronecker incidentally denied 
the "existence" of TT, and he was less of a radical 
at that than some of the moderns. 

What is the point at issue here? There are sev- 
eral. One which is disturbing mathematicians 
profoundly at present is this very question of what 
is meant by mathematical existence. We know 
or used to think we knew that with sufficient dili- 


gence (and stupidity) TT = 3.1415926 could 

be calculated to an indefinitely great number of 
decimals. Indefinitely great? Not exactly; for 
who could ever do it? In what sense then, if any, 
does TT "exist" as an infinite, nonrepeating deci- 
mal? I trust that I have not made this sound like 
a foolish quibble, for it is anything but that. 

Kronecker said flatly that unless we can give a 
definite means of constructing the mathematical 
things about which we talk and think we are reason- 
ing, we are talking nonsense and not reasoning at all. 
At one stroke he denied the validity of all the great 
work of the mathematical analysts on the infinite. 
To him it was worse than meaningless; it was use- 

There are those today who say Kronecker was 
right, and they cannot be silenced by an affection 
of superiority on the part of those who believe 
otherwise. There are equally strong men on both 
sides of the entire controversy. 

Progress in this direction is being made by meet- 
ing Kronecker's objection step by step where it is 
important to do so, and actually exhibiting con- 
structions for things that are used. It is impos- 
sible, of course, to meet fully any demand for a 
construction of an actual infinite; here we have to 
be content with exhibiting a process which, if 
carried out, would produce the required thing to 
any prescribed degree of accuracy. 


I must warn the reader that the foregoing is an 
exceedingly crude description of extremely subtle 
difficulties, and that parts of the last sentence, if 
not all of it, would be regarded as sheer nonsense 
by one of the modern schools of mathematical 
thought. I can only suggest those profound prob- 
lems and pass on to others, treating them equally 

The use of words alone in all these discussions is 
a treacherous proceeding. This also affects much 
of the technical literature on these disputes. 
Some mathematicians feel that if the ideas con- 
sidered can not be adequately expressed in some 
appropriate symbolism, they are too dangerous to 
be handled. The history of philosophy is a suffi- 
cient warning. 


So great is the average mathematician's distrust 
of purely verbal arguments that Hilbert, beginning 
about 1925, proposed that for the present at least 
mathematicians forget about the "meanings" of 
their elaborate game with symbols, and concen- 
trate their attention on the game itself. He and 
his pupils have formulated the rules of play in an 
unassuming theory of demonstration, whose aim is 
to prove that mathematics is free of contradiction. 

The rules are expressed in symbols with brief 


verbal instructions for their use, and are a strik- 
ingly simple form of symbolic logic. Hilbert 
assumes as known the logical and, or, not, if-then, 
and some more mathematical notions, equally 
elementary. For example, not-X is written X; 
X & Y is X and Y, etc., and a typical rule of play 
permits us to put JT&F for Y&X if the latter 
appears in shifting the "meaningless marks" about 
in accordance with the rules. 

The object of all this is to prove that the innocent 
looking rules will never lead to a contradiction. 

Critics of the movement deny that it has any 
significance for the points in dispute. Some of 
them further deny the modest claim made by 
Hilbert's adherents that they have proved the 
consistency of mathematics up to the point where 
only Si finite number, or only a countable infinity of 
elements are concerned. A countable infinity, we 
recall, is as many as there are of all the natural 
numbers 1, 2, 3, That wretched mono- 
syllable "all" has caused mathematicians more 
trouble than all the rest of the dictionary. 

That Hilbert's method has established the con- 
sistency of the Dedekind cut, of Cantor's theory of 
the infinite, of any of the theory of sets, or of 
mathematical analysis, is not claimed even by its 
most ardent partisans. 

The outstanding merit of Hilbert's contribution 


is its fearless exposure of the weak spots in mathe- 
matics and the attention which it commands for 
these spots from competent professional ma f he- 
maticians. That a man of Hilbert's mathematical 
eminence should put the supreme eff ort of his great 
career into this crisis is a sufficient reply to those 
who belittle the honest if disturbing strictures of 
the critics. 


Most of the paradoxes which mathematics is 
striving to resolve entered with the mathematical 
infinite. This led to a critical examination of 
mathematical reasoning of any type. From this 
the criticisms have reached out to the classical 
logic of Aristotle, which for over two thousand 
years reigned free of suspicion that it might not be 
universally valid. 

If Aristotle ever heard of an infinite set in the 
sense of mathematics, he seems to have left no 
record of the fact. What reason can be given that 
Aristotelian logic applied to infinite sets will not 
produce contradictions? None whatever. In 
1906 Henri Lebesgue, who revolutionized the 
theory of integration, stated explicitly that he was 
not convinced that a statement about an infinite 
set is necessarily true or false. In other words 
there may be a third possibility, between truth and 


falsity, or there may be nothing but nonsense in 
any assertion about an infinite set. 

The reader may amuse himself by picking the 
foregoing sentence to pieces in the light of the very 
objection it raises, namely to the universal validity 
of the law of excluded middle an assertion is true 
or it is false. The sentence is riddled with incon- 
sistencies. It is a fair sample of the difficulty of 
talking sense about the fundamentals of reason- 
ing mathematical or other. 

A leading contention of the strong intuitionist 
school led by L. E. J. Brouwer is that the logic of 
Aristotle is partly inappropriate for mathematics. 
In particular, the law of excluded middle is not 
always admissible, and Euclid's method of indirect 
proof is not free from very serious objection. One 
aim of this school is to revise mathematical reason- 
ing so as to avoid the disputed points. It is aston- 
ishing to see how far we can go in this direction. 

What is desired in this: to weed out what can 
be shown to be definitely erroneous in mathe- 
matics, and to root what remains in uninfected 
soil. Further, if the whole critical movement is 
not to be utterly barren, it must account for the 
undisputed fact that mathematical reasoning has 
led to results which, by common consent, are 
true, whatever "truth" may mean. If the reason- 


ing by which correct results were reached was 
irreparably wrong, then that in itself will be an 
astounding and far-reaching discovery. 

As the reader may be interested in seeing the 
kind of puzzle which is worrying mathematicians, 
I shall transcribe a simple one for him to ponder. 
It is known as Russell's paradox. This particular 
one is cited because Russell with A. N. Whitehead 
in 1910-1914 produced the monumental Principia 
Mathematica which aimed, among other things, 
to resolve the paradoxes of analysis and the theory 
of sets. This gave a new impulse to mathematical 
rigor which has lasted to the present day. Here is 
the paradox, not yet resolved, as the machinery 
which Russell contructed for such purposes has 
been abandoned by mathematicians. 

"Let w be the class of all those classes which are 
not members of themselves. Then, whatever 
class x may be, x is a w is equivalent to x is not an 
x. Hence, giving to x the value w 9 6 w is a w' is 
equivalent to 'w is not a w\ 

Two propositions are called equivalent when both 
are true or both are false." (American Journal of 
Mathematics, vol. 30, 1908, p. 222.) 

It was the appearance of several similar para- 
doxes in mathematical analysis in the past forty 
years that led to the present upheaval. 



After the splendid achievements of the Century 
of Progress in mathematics, it seems ungracious 
to close on a note of doubt. The sentiments of 
creative mathematicians cannot be disregarded. 
Surely their feeling for what is true in mathe- 
matics is not without significance. Almost with- 
out exception these men feel this about the past 
and probable future of their beloved mathematics: 
not all of those giants of the past can have been 
fools all of the time, and we may rest assured that 
greater shall come after them. 

Wisdom was not born with us, nor will it perish 
when we descend into the shadows with a regretful 
backward glance that other eyes than ours are 
already lit by the dawn of a new and truer mathe-